Etale Homotopy of Simplicial Schemes. (AM-104), Volume 104 9781400881499

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Etale Homotopy of Simplicial Schemes

INTRODUCTION In the early part of this century, alg eb raic geometry and alg eb raic topology were not se p a ra te d isc ip lin e s . Indeed many manifolds admit the structure of a (com plex) alg eb raic v ariety .

S. L e fs c h e tz and others intro­

duced algeb ro-geom etric techniques to study to p o lo g ical properties of such v a rie tie s. Subsequently, alg eb raic geometry widened its sco p e to include v a rie tie s of p ositiv e c h a ra c te r is tic , rings arisin g in alg eb raic number theory, and more general sch em es. An alg eb rizatio n of alg eb raic geometry has been ach iev ed , thanks in large part to A. Weil and 0 . Z a risk i; and powerful techniques of hom ological algebra have been employed, e s p e cia lly th ose introduced by J . - P . Serre and A. G rothendieck. On the other hand, alg eb raic topology has developed in d irectio n s not obviously relevant to alg eb raic geometry. Inspired by A. W eil's celeb rated co n je ctu re s, M. Artin and A. Grothen­ d ieck developed e ta le cohomology theory, a theory su fficien tly well behaved that in some w ays it plays the role for p o sitiv e c h a ra c te ris tic v a rie tie s of singular cohomology theory in alg eb ra ic topology.

P . D elig n e’s

proof of the Weil C onjectu res [22] is a dram atic application of e ta le cohom ology.

M. Artin and B . Mazur refined e ta le cohomology theory in

their book entitled E ta le Homotopy [8]:

they introduced a “ pro-homotopy

ty p e " a s s o c ia te d to a schem e whose cohomology is the e ta le cohomology of the sch em e. T hu s, e ta le homotopy theory reinforces the relationship betw een algeb raic geometry and alg eb raic topology. The purpose of th is book is to provide a coh eren t acco u n t of the current s ta te of e ta le homotopy theory. We d escrib e in C hapters 9 , 1 1 , and 1 2 , various ap p licatio n s of this theory to alg eb raic topology, cohomology of groups, and alg eb raic geom etry. T h e se ap p licatio n s have required 3

4

E T A L E HOMOTOPY O F SIM PLIC IA L SCHEMES

repeated refinement and g en eralizatio n of the theory developed by M. Artin and B . Mazur. F o r this reaso n , we con sid er sim p licial sch em es throughout, n e ce ssa ry for alm ost a ll e x istin g ap p lica tio n s, and introduce the e ta le top o lo g ical type which refines the e ta le homotopy type. With an eye to future a p p lica tio n s, we introduce various alg eb raic in­ varian ts of the e ta le top o lo g ical type not usually considered by alg eb raic geom eters:

function co m p lexes, re la tiv e homology and cohom ology, and

generalized cohom ology. It is our hope that this p resentation of e ta le homotopy theory w ill enable ap p licatio n s by to p o lo g ists and geom eters who have not immersed th em selves in the d e lic a te te c h n ic a litie s of the su b je ct. C onsequently, we have attem pted to minimize p re-requ isites to b a sic hom ological algebra [1 7 ], elem entary theory of sim p licial s e ts [57], and an acq u ain tan ce with alg eb raic geometry [50]. We have taken th is opportunity to re-work much of the foundational m aterial of e ta le homotopy developed by M. A rtin-B . Mazur and the author:

the knowledgeable reader

will reco g n ize many improvements and g en eralizatio n s of resu lts in the literatu re. We now proceed to briefly sk etch the con ten ts and organization of the various ch a p te rs, e a ch of which a lso has its own introduction. Beginning with the definition of a sim p licial sch em e, Chapter 1 d e scrib e s exam ples most common in ap p licatio n s and briefly d is c u s s e s a technique for ad apt­ ing co n stru ctio n s on sch em es to apply to sim p licial sch em es. Chapter 2 defines (e ta le ) sh eaf cohomology in terms of derived functors and re la te s the sh eaf cohomology of a sim p licial schem e to that of its con stituen t sch em es. A fter d iscu ssin g the somewhat more fam iliar C ech co n stru ctio n , we d escrib e in Chapter 3 the analogous con stru ctio n of J .- L . Verdier which en ables one to compute cohomology in terms of h ypercoverings. Chapter 4 p resen ts the e ta le to p o lo g ical type of a sim p licial schem e based on the in verse system of rigid hypercoverings (in itially inspired by work of S. Lubkin [54]). T his e ta le to p o lo g ical type is a refinement of the Artin-Mazur e ta le homotopy type and is given by a somewhat more natural co n stru ctio n than previous refinem ents. In Chapter 5 , we verify that the

5

INTRODUCTION

s e t of con n ected com ponents, the fundamental group, and the cohomology with lo cal co e fficie n ts of the e ta le to p o lo g ical type of a sim p licial schem e are given by the s e t of co n nected com ponents, the Grothendieck fundamen­ tal group, and the sh eaf cohomology of the sim p licial schem e. B e c a u s e the e ta le top o lo g ical type is not a sin g le sim p licial s e t but an in verse sy stem , m achinery must be employed to an aly ze its homotopy in­ varian ts. Chapter 6 com pares co n stru ctio n s of M. A rtin-B . Mazur, D. Sullivan, and A. K. B ousfield -D . M. Kan, e a ch of which has been em­ ployed in the literatu re for various a p p lica tio n s. To a s s i s t in the id en tifi­ catio n of various homotopy invariants using th e se co n stru ctio n s, we prove se v e ra l fin iten ess properties in Chapter 7. Chapter 8 then re la te s the e ta le top ological type of a sim p licial com plex variety to the geom etric realizatio n of its a s s o c ia te d sim p licial (a n a ly tic ) s p a c e and provides a com parison of top o lo gical types in c h a ra c te r is tic

p ^ 0 and

0.

An e sp e cia lly important

exam ple is that of the cla ssify in g s p a c e of a com plex red u ctive L ie group. Having travelled so far, the reader is rewarded with various a p p lica ­ tio n s. In terest in e ta le homotopy theory w as aroused by D. Q uillen’s use of the theory in his sk e tch of a proof of the (com plex) Adams co n jectu re [61].

We present a m odification of D. S u llivan ’s proof of the Adams co n ­

jectu re [69] as well a s our proof of its infinite loop sp a c e refinement [37]. Chapter 9 a lso d e scrib e s the au th or’s use of homomorphisms of alg eb raic groups in p ositive c h a ra c te r is tic to provide in terestin g maps of c la s s ify in g s p a c e s and homogeneous s p a c e s of com p act L ie groups [33], [36]. Chapter 11 p resen ts four geom etric ap p licatio n s of e ta le homotopy theory originally due to D eligne-Sullivan [25], D. Cox [20], and the author [3 0 ], [31].

The

la s t two ap p licatio n s of Chapter 11 and all th ose of Chapter 12 require a com parison of homotopy types of the algeb ro-geom etric and homotopy fibres a s d iscu sse d in Chapter 1 0 .

In Chapter 1 2 , we d escrib e further

ap p lication s by the author of e ta le homotopy to K -theories of finite field s and the cohomology of finite C hevalley groups [32], [34], [35], [41]. The d iscu ssio n of function com p lexes in Chapter 13 is cen tral to recen t ap p licatio n s of e ta le homotopy to alg eb raic K-theory by the author

6

E T A L E HOMOTOPY O F SIM PLIC IA L SCHEMES

and W. Dwyer [27], [39].

B e c a u s e this is the first published acco u n t of

function com p lexes, Chapter 13 exam ines their behavior in some d e ta il. In Chapter 1 4 , we d is c u s s re lativ e cohomology in order to incorporate lo c a l cohomology and cohomology with supports in the framework of e ta le homotopy theory. Chapter 15 provides a refinement of D. C o x ’s co n s tru c ­ tion of tubular neighborhoods [18] and u tiliz e s th e se tubular neighborhoods in a d iscu ssio n of e x c is io n and M ayer-V ietoris. Chapter 16 provides a first introduction of generalized cohomology into the study of alg eb raic geom etry: an early v ersion of this chapter was the b a sis for the au th or’s com parison of alg eb raic and to p o lo g ical K-theory of v a rie tie s [38].

F in a lly , Chapter

17 p resen ts a sk etch of P oin care' duality for e ta le cohomology using the machinery developed in the preceding ch a p te rs. During the lengthy evolution of th is book, further developm ents and ap p lication s of e ta le homotopy theory have a rise n . We refer the reader to [73] for a d iscu ssio n of sh e a v e s of s p e ctra and e ta le cohom ological d escen t u tilized in a further ap plication to a lg eb raic K-theory. We are e s p e c ia lly indebted to D. C ox, who shared with us pre-prints of his work on the homotopy type of sim p licial sch em es [19] and on tubular neighborhoods [18].

We a lso thank A. K. B o u sfield , W. Dwyer, and

R . Thom ason for many valuable co n v e rsa tio n s. F in a lly , we thank Oxford U n iversity, the U niversity of Cambridge, and the Institute for Advanced Study for their warm h osp itality during the writing of this book, and g rate­ fully acknow ledge support from the N ational S cie n ce Foundation and the S cien tific R e se a rch C ouncil.

1.

E T A L E SITE OF A SIMPLICIAL SCHEME

After statin g the definition of a sim p licial sch em e, we provide s e v e ra l important exam ples in E xam p les 1 .1 , 1 .2 , and 1 .3 .

The first two exam ples

a rise frequently in ap p lica tio n s, w hereas the third is cen tral in the co n ­ struction of the e ta le homotopy type. e ta le site of a sim p licial sch em e.

In D efinition 1 .4 , we introduce the

As we exp lain , the e ta le s ite is a

gen eralization due to G rothendieck of the categ o ry of open su b se ts of a top olo gical s p a c e . We conclude this chapter with a co n stru ctio n which en ables one to extend ce rta in arguments applied to a given dimension of a sim p licial sch em e to the entire sim p licial sch em e. We re ca ll th at A , the ca teg o ry of standard s im p lic e s , c o n s is ts of o b jects

A[n] for n > 0 and maps

function from | 0 ,l,---,n ! to

A[n] -> A[m] for e a ch nondecreasing

{ 0 , 1 , •••,m|. Any such map a : A [ n ] A [ m ]

(other than the id entity) can be written as a com posite of “ d e g e n e ra cy ” maps

A[k—1 ] with j < k -1

o su rje ctiv e and “ f a c e ” maps

defined by

A [£+l] with i < £+1 defined as

that in jectiv e map su ch that i e {0,•••,£+! \ is not in the image of d- . A sim p licia l o b ject of a categ o ry opposite categ o ry to

A;

C is a functor A 0 -» C ,

where A 0 is the

a map of sim p licial o b jects is a natural tra n s­

formation of functors. A sim p licia l s c h e m e is a functor from A 0 to the categ o ry of sch em es ( = lo ca l ringed s p a c e s which admit a covering by open su b sp a ce s isom or­ phic to the sp ectra of rings; c f ., [50], II.2).

Follow ing conventional n o ta­

tion, we denote such a sim p licial sch em e by X . , and we denote X .(A [n ]) by X n . E X A M P L E 1 .1 . X®T.

Let

X

be a schem e and

is the sim p licial schem e with

T.

a sim p licial s e t.

Then

( X ® T .) n equal to the disjoint union 7

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

8

indexed by T n and with a : ( X ® T .) m -> (X ® T .)n for

of co p ies of X

a : A[n] -» A[m] in A given by sending the summand of X

indexed by

t e T m via the identity to the summand of X

indexed by a (t) e T n . More

gen erally, if X .

T.

X .® T .

is a sim p licial schem e and

a sim p licial s e t, then

is sim ilarly defined with (X . ® T .)n equal to the d isjoin t union of

co p ies of X n indexed by T n . Let

A[k] a ls o denote the sim p licial s e t defined by A [k]n =

H om ^(A[n], A [k]) . In p articu lar, is equal to X

in e a ch dimension (we sh a ll usually denote

simply by X ). X ® A[k] -> Y .

X ® A [0 ] is the sim p licial schem e which X ® A [0]

We e a sily verify that a map of sim p licia l sch em es

is eq uivalent to a map of sch em es

X

. A sp e c ia l c a s e

of Exam ple 1.1 which we sh all frequently employ is that in which T. equals

A [1 ] s o that

(X .® A [ l ] ) n is the d isjoin t union of n+1

X n . Two maps f,g : X . -> Y .

of sim p licial sch em es are said to be related

by the sim p licia l homotopy H : X .® A [l] -»Y . tions of H to X .® A [0 ] are If X

f and

if the two can o n ical re s tric ­

g.

is a schem e which appears a c y c lic with re sp e ct to a ce rta in

cohomology theory (e .g .,

X might be the spectrum of an a lg eb raically

clo sed field ), then X ® T . c ia l s e t

co p ies of

T.

appears to be an alg eb raic model of the sim pli­

for this cohomology theory.

T his is exploited in Chapter 11

to study finite C hevalley groups, as well a s by Z . Wojtkowiak in [72], and (im p licitly) by C. Soule in [67].

U sing the s p e c ia l c a s e

T = B Z /2 ,

D. Cox has studied real alg eb raic v a rie tie s (s e e C orollary 1 1 .3 ). ■ S be a sch em e.

A group s c h e m e over S , G ,

E X A M P L E 1 .2 .

Let

is a

map of sch em es

G -» S with the property that Homs ( ,G ) is a group

valued functor on the categ o ry of sch em es over S (i .e ., maps of sch em es Z -> S ).

In other words,

p : G xs G ^G

G is provided with maps

over S sa tisfy in g the usual axiom s.

e : S -> G and If G is a group

schem e over S , the c la s s ify in g s im p licia l s c h e m e BG is the sim p licial schem e given in dimension

n > 0 by (®G )n = Gx n , the n-fold fibre

product of G with itse lf over S , and with stru ctu re maps given in the

9

1. E T A L E S IT E OF A SIM PLIC IA L SCHEME

usual way using e

and

/z.

More gen erally, if X

schem e over S provided with left G -action G -action

(re sp e ctiv e ly ,

a: G xg X ^ X

Y ) is a

(re s p ., right

/3 : Y x g G -> G ), then B (Y ,G ,X ) is the sim p licial schem e given

in dimension n > 0 by B (Y ,G ,X )n = Y x s Gx n x s X and with stru ctu re maps given in the usual way using e , [jl , a and

/3

(c f ., [58], for an e x p licit d escrip tion of this “ d o u b le bar co n stru ctio n ” ). We let B (Y ,G ,* ) (re s p .,

B (* ,G ,X ) ) be the sim p licia l sch em es obtained

from B (Y ,G ,X ) by deleting the right-hand (re sp ., left-hand) facto r. S pecial c a s e s of Exam ple 1 .2 have occurred in most of the a p p lica ­ tions of e ta le homotopy theory. T h ese exam ples are so prevalent b e ca u se any com plex red u ctive L ie group G (C) adm its the structure of an a lg e ­ b raic group G^, over Spec C; BG^, then provides an alg eb raic model for the top ological c la ss ify in g sp a ce of G (C ). Moreover, if H is a subgroup sch em e of G over S , then H a c ts on G by m ultiplication so that B (G ,H ,*) and B (*,H ,G ) se rv e a s models for the “ homogeneous s p a c e ” G /H

(se e Chapter 9 ). ■

E X A M P L E 1 .3.

Let

For

x / n ^ denote the n-th truncation of X . : x / n ^ is the

n > 0,

let

restrictio n of X . F o r any

n > 0,

(or c o s k nX .

X.

be a sim p licial schem e over a given schem e

to the fu ll-su bcategory of A0 with o b jects we define the n-th c o s k eleto n of X .

S.

A[k], k < n .

over S, co sk ^ X .

if no confusion a ris e s by leaving S im plicit) by the follow ­

ing universal property:

F o r any sim p licial sch em e

of maps Y . -> co sk ^ X .

over S is in natural one-to-one correspon d ence

with the s e t of maps Y / n^ -> x / n ^ over S .

Y.

over S ,

the se t

If we view co sk ^ ( ) as a

functor from n-truncated sim p licial sch em es to sim p licial sch e m e s, then cosk ^ ( If X

) is right adjoint to the n-truncation functor ( )^n\ is a schem e over S, cosk ^ X

n erv e (u su ally w ritten Ng(X ) ) of X

( = co sk ^ (X ® A [0]) ) is the C e c h over S with

(co sk g X )n equal to

the (n-fl)-fold fibre product of X with its e lf over S . One v erifies that

10

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

co sk ^ X . k S p e c k ,

and a sh eaf F

of ab elian groups on E t(S p ec k) is

equivalent to the data of the group co lim F (S p ec K -►Spec k) = F (k ) provided K /k with a left actio n of the galois group lizer of any elem ent of F (k )

lim G a l(K /k ) such that the sta b iK /k

is a subgroup of finite index ([5 9 ], II. 1 .9 ).

In p articu lar, if k is a separably (a lg e b ra ica lly ) clo se d field, than a sh eaf F

on E t(S p ec k) is equivalent to its value F (S p e c k ) . Viewing

such S p eck

as “ p o in ts” for the e ta le topology, we define a g eo m etric

point of a sim p licial schem e X .

to be a map Spec k->Xm ,

some

m> 0

2. SH EA VES AND COHOMOLOGY

and some sep arab ly clo se d field

k.

15

We define the sta lk of a sh eaf F

on

E t(X .) at the geom etric point a : S p e ck -> X m to be a * F m(S p e c k ). We proceed to the definition of the (e ta le ) cohomology of an a b elia n s h e a f ( i .e ., a sh e a f of ab elian groups) F

on E t ( X .)

using derived

fu n ctors. F o r a more detailed d is cu ssio n , we refer the reader to [59]. L et X.

PROPOSITION 2 .2 .

be a s im p licia l s c h e m e , and let AbSh(X.)

den o te the ca tego ry of a b elia n s h e a v e s on E t ( X .) .

T h en A bSh(X .)

is an

a b elian ca tego ry with enough in je c tiv e s . M oreover, a s e q u e n c e of s h e a v e s in A bSh(X .) X m of X .

is e x a c t if and only if for ev ery g eo m etric point a : Spec k -*

the s e q u e n c e of sta lk s at a

P ro o f. A map A cokernel

is ex a ct.

in A bSh(X .) has kernel

B -> C ) if and only if the re strictio n

K^A

(re sp e ctiv e ly ,

Kn -> An (re s p .,

B n ^ C n)

is the kernel (re sp ., cokernel) of A n -» B n in AbSh(Xn) for e a ch

n > 0,

where AbSh(Xn) is the ab elian categ o ry of ab elian s h e a v e s on E t(X n) (cf. [59], II.2 .1 5 ). categ o ry .

T his readily im plies that A bSh(X.) is an ab elian

T o show

A bSh(X .) has enough in je c tiv e s , we use the functor

R n( ) : AbSh(Xn) -> A bSh(X .) defined by sending G e AbSh(Xn) to R n(G) with (R „(G ))rn = n m

a ^. G .

II

B ecau se

R IIn( ) is right adjoint to the

A[nL restrictio n functor (which is e x a c t),

R n(G) is in je ctiv e in A bSh(X.)

whenever G is in je ctiv e in AbSh(Xn) . T hu s, if F

is an arbitrary

abelian sh eaf on E t(X .) and if Fn -> I in jectiv e

In of AbSh(Xn) for ea ch

monomorphism of F

is a monomorphism of F n in an CO n > 0 , then F -» II R n(In) is a

into an in jectiv e of A b S h (X .).

n=0

B e c a u s e a seq uen ce

of sh eav es in A bSh(X .) is e x a c t if and only if its re strictio n to e a ch AbSh(Xn) is e x a c t, the la s t statem ent follows from the corresponding fact for AbSh(Xn) (cf. [5 9 ], I I .2 .1 5 ). ■ The following definition of the (e ta le ) cohomology of a sim p licial schem e (due to P . D eligne in [23], 5 .2 ) is the natural gen eralization of the

16

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

definition of cohomology of a schem e as derived functors of the global se ctio n functor. DEFINITION 2 .3 .

Let X.

be a sim p licial sch em e.

F o r any

i>0,

the

cohom ology group functor Hi (X. , ) : A bSh(X .) -> Ab is the i-th right derived functor of the functor sending an abelian sh e a f F on E t( X .)

to the ab elian group given a s the kernel of the map

d * - d * : F ( X 0) - ^ ( X ^ .

E q u iv alen tly ,

H i(X . , ) = E x t ^ b sh (x> )(Z , where to

)

Z is the co n stan t ab elian sh eaf on E t(X .) with e a ch sta lk equal

Z. ■ The definition of H *(X . ,F )

is global in the s e n se th at it is given in

terms of E t(X .) rather than the various

E t ( X n) .

N on eth eless, the follow ­

ing proposition en ables us to re la te H *(X . , F ) to the various H *(X n,F n), n > 0 . PROPOSITION 2 .4 . F e A b S h (X .).

L et X.

b e a sim p licia l s c h e m e , and let

T h en there e x is t s a first quadrant s p e ctra l s e q u e n c e E f ’t = ^ ( X g .F g ) => Hs + t(X . ,F )

natural with r e s p e c t to X . Proof.

Let

and F .

F -> I’ be an in jectiv e resolution in A b S h (X .).

We con sid er

the functor L n( ) : AbSh(Xn) -> A bSh(X .) defined by sending G e AbSh(Xn) to L n(G) defined by ( L n(G ))m =

© a*G , any n > 0 . aeA[m]n

B ecau se

L n( ) is an e x a c t left adjoint to the re strictio n functor ( ) n , we co n ­ clude that

( )

sends in je ctiv es to in je ctiv e s for e a ch

Fn -» (I*)n is an in jectiv e resolution in AbSh(Xn) .

n > 0.

T hus,

17

2. SH EAVES AND COHOMOLOGY

Let

Zx

denote the com plex of ab elian sh e a v e s in A bSh(X.) * whose m-th term is the ab elian sh eaf represented (in the sm all) by id : X m -> X m in E t ( X m) (and whose differen tial is the usual alternating sum of the maps induced by d- : X m -> X m_ 1 , i < m ). L m(Z|x ) ; when re stricte d to ' m

Zx

-> Z by defining Z Q(U) = *

Z.

We define an augmentation

© Z (U ) -> Z(U ) for any A [n] 0

in E t( X .) to be the identity on e ach summand. is e x a c t in A b S h (X .),

Z m equals

E t(X n) , Z m is the d irect sum indexed by

A[n]m of copies of the con stan t sh eaf map

T hus,

To verify that

U -» X n Zx

-> Z->0

*

it su ffice s to verify that the re strictio n s

(Z x )n Z -> 0 are e x a c t in AbSh(Xn) for n > 0 . This e x a c tn e s s follows * from the a c y c li c ity of A[n] and the identification of ( Z x ) n -> Z with Z® (A[n] -> A [ 0 ] ) . The sp e ctra l seq uen ce is obtained from the bi-com plex ^ omAbSh(X )( ^ x H °mA bsh(X

to tal cohomology of the bi-com plex is that of which is

H *(X . , F ) ;

w hereas the cohomology of

H° mA b S h ( X . ) ( V n = Ho mAb s h (x m)(Z >(Om) is

H *(X m, F m) . -

As an immediate corollary of P rop osition 2 .4 , we conclude that if F c A bSh(X .) is su ch that F n e AbSh(Xn) is in je ctiv e for e ach then F

is a c y c lic ( i .e .,

co ch ain com plex

n > 0,

H \ X . ,F ) = 0 for i > 0 ) if and only if the

F ( X .) = In h> F ( X n)i is a c y c li c .

We sh all often u tilize b i-sim p licial sch em es to study sim p licial sch em es.

As the reader can s e e from the following brief d iscu ssio n , the

e tale topology and e ta le cohomology of b i-sim p licial sch em es are defined analogously to that of sim p licial sch e m e s.

Moreover, P rop osition 2 .5

permits us to rep la ce a b i-sim p licial schem e c ia l sc h e m e A X .,

X ..

by its diagonal sim pli-

(where (A X ..) n = X n n) .

A b i-sim p licia l s c h e m e is a functor from A0 x A0 to the categ o ry of sch em es.

Follow in g convention, we denote su ch a b i-sim p licial schem e

by X .. , and we denote X ..(A [m ], A[n]) by X m n . We let

E t ( X ..)

denote

18

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

the category whose ob jects a r e retale maps

U

X g ^ for some s , t > 0

and whose maps are commutative squares U

►V

X s , t ---------------------" x k,£ whose bottom arrow is a specified structure map of X .. . We give

E t(X ..)

the etale topology by defining a covering of U -> X c f to be a family of over X g ^ whose images cover

etale maps

a lso denote the resultant e ta le site on the category

U.

We let E t ( X . . )

E t(X ..) .

We define A bSh(X..) to be the abelian category of abelian s h e a v e s on the s it e

E t(X ..),

where the sh eaf axiom is that of Definition 2 . 1.

(e ta le ) cohom ology groups of a b i-sim p licia l s c h e m e X . . sheaf F e AbSh(X..)

The

with values in a

are defined to be

H \ X . , F ) = E < bsh(x> )( Z , F ) where

Z is the con stan t abelian sh eaf on E t ( X . ) with fibres

PROPOSITION 2 . 5 .

L e t X ..

be a b i-sim p licia l s c h e m e .

Z.

T h en th ere is a

natural isom orphism of 8 -functors H *(X .. , ) ^ on A b S h (X ..), and (

w here A X .,

H *(A X .. ,(

)A)

is the diagonal (s im p licia l sch e m e ) of X . . ,

: AbS h (X ..) -» AbSh(AX..)

is the restrictio n functor.

P roof. It su ffices to (a ) exhibit a natural isomorphism H ° ( X .. , ) H ° ( A X .. ,(

)A) ,

(b) prove H ^ A X .. ,IA) = 0 for i > 0

in je ctive , and (c) verify that H *(A X .. ,(

any

I e AbSh(X ..)

)A) is in fact a 8 -fu n cto r (i .e .,

there is naturally a s s o c ia t e d a long e x a c t seq uen ce

19

2. SH EA VES AND COHOMOLOGY

••• -> H '(A X .. , F A) -> Hi(A X .. ,G A) -> H ^ A X .. ,HA) -> Hi+1(A X .. , F A) - ••• to each short e x a c t seq u en ce V. 4 .4 ).

0^F^G->H->0

in A b S h (X ..)) (c f ., [17],

P a rt (a ) is readily ach ieved by identifying both H ° ( X ..,

) and

H °(A X .. ,( ) A) with the group K e r ( d £ ° d * - d * ° d * : F ( X 0 0) - F ( X 1(1) ) . Arguing as in the proof of Prop osition 2 .4 , we verify that is in jectiv e whenever I e A b S h (X ..) is in je ctiv e . su ffice s to prove that

I^ (A X ..) == {n k I(X n

I e A b S h (X ..) is in je ctiv e .

B ecau se

)}

Ig t e AbSh(Xg t)

Thus, to prove (b), it

is a c y c lic whenever

T o t(I (X ..)) and A l(X ..) = IA(A X ..)

have the sam e cohomology ([2 6 ], 2 .9 ), this follows from the e x a c tn e s s of T o t(Z x

) whose proof is analogous to that of the e x a c tn e s s of

given in P rop osition 2 .4 . ( )^

is e x a c t b e ca u se

only if 0

F g t ^ Gg t

Zx

F in a lly , to prove (c ) , it su ffice s to observe that 0^F-*G-*H-*0

is e x a c t in A b S h (X ..) if and

Hg t ^ 0 is e x a c t in AbSh(Xg t) for all

s ,t > 0 . ■ We conclude this ch apter with the following b i-sim p licial analogue of P roposition 2 .4 . PROPOSITION 2 .6 .

L e t X ..

a b elia n s h e a f on E t ( X . . ) .

b e a b i-sim p licia l s c h e m e and let F

b e an

T h en there e x is t s a first quadrant s p e c tra l

sequ en ce E f ' 1 = H ^ X ^ ,F Sm) = > Hs + t(X .. ,F ) natural with r e s p e c t to X ..

and F .

P ro o f. Em ploying L s * ( ) : AbSh(Xg )-> A b S h (X ..) as

in the proof of

P rop osition 2 .4 , we conclude that Ig e AbSh(Xg ) is in jectiv e whenever I 6 AbSh(X..)

is in je ctiv e .

We define the com plex Z x of sh e a v e s on

E t ( X ..) with n-th term equal to L n* (Z ) e A b S h (X ..) , so th at Z Y is a resolution in A b S h (X ..). a s s o c ia te d to the bicom plex

-^Z

The a s se rte d s p e ctra l seq u en ce is that H °mAbsh(X

an in jectiv e resolution in A b S h (X ..). a

)( ^ x

where F -> I* is

3.

COHOMOLOGY VIA HYPERCOVERINGS

The main resu lt of this ch apter is Theorem 3 .8 which a s s e rts that sh eaf cohomology of a sim p licial schem e ca n be computed in a som ew hat com binatorial way using h ypercoverin gs. B e c a u s e the definition of a hyper­ covering and the required properties of the categ o ry of hypercoverings are somewhat formidable at first encounter, we begin this ch apter with a d iscu ssio n of the sim pler co n te x t of C ech nerves and C ech cohom ology.

As

seen in C orollary 3 .9 , C ech cohomology is naturally isom orphic to sh eaf cohomology in many c a s e s of in terest (this is a very s p e c ia l property of the e ta le s ite ).

By considering h ypercoverin gs, we provide com binatorial

arguments ap p licab le to any s ite only notationally more com plicated than th ose of C ech theory (s e e , for exam ple, Lemma 8 .3 ). be a sim p licial sch em e. An (e ta le ) co v erin g U. -> X . of X . \/ is an e tale su rje ctiv e map. The C e c h n e rv e of U. -> X . is the b isim p licial Let

schem e

X.

Nx (U .) defined by

Nx . ( u " ) s ,t = (N x s (Us ))t the (t+ l)-fold fibre product of U s with its e lf over X g (s e e Exam ple 1 .3 ). F o r any a b elia n p re s h e a f P define s ,t

( i .e ., functor E t ( X .) °

(a b .g rps) ), we

P (N X (U .)) to be the b i-coch ain com plex given in bi-codim ension

by P (N X (U ..)S f.) with d ifferen tials obtained in the usual way a s an

altern atin g sum of the maps

d * . We re c a ll that the cohomology of su ch a

bi-com plex is the cohomology of the a s s o c ia te d to tal com plex b e a sim p licia l s c h e m e and P an a b elia n \/ s h e a f on X . . For any i > 0 , d e fin e the C e c h cohom ology of X . with

PROPOSITION 3 .1 .

v a lu es in P

L et X.

in d e g re e i by

20

3. COHOMOLOGY VIA H YPERC O V ER IN G S

21

= colim H^PCNy (U .)))

u.^x.

A*

w here the colim it is in d ex e d by c o v erin g s U. -> X . of X . So d efin e d , v* H (X , ) is a 8 -functor on the ca teg o ry of a b elia n p re s h e a v e s on X . . P ro o f. L e t

U. -» X .

and V. -> X .

be co v erin g s of X .

defined by (U. x x V .)n = ( ^ n x x more, if f,g : U. -> V.

Then

*s a ^so a covering of X .

colim

F u rth er­

are two maps over X . , then

Nx . ( f )* = Nx . ( g ) * : H *(P (N X (V .))) - H *(P (N X>(U .))) T hus,

U. x x V. ->X.

(s e e below).

can be re-indexed by the opposite categ o ry to the left

U.->X.

d ir e c te d ca tego ry ( i .e ., category with the properties that there is at most one map between any two o b je cts and th at for any two o b jects there e x is ts a third mapping to both) of coverin gs of X . maps, and h ence is e x a c t. f,g : U. -> V.

and eq uivalen ce c l a s s e s of

The fa c t that Nx (f )* = Nx (g )*

is proved using the fa ct that two maps

for

fn,g n ' Un -> Vn over

X n for n > 0 have the property that

c o s k 0 n(fn), c o s k 0 n(g n) : c o s k 0 n(Un) -> c o s k Qn(Vn) X

are related by a unique sim p licial homotopy (sin ce a map c o s k Q (Un) ® A [1 ]

xn

-> c o s k Q (Vfl)

is equivalent to its re strictio n to dimension

0 ).

T hu s,

Nx ( f ) and Nx (g) are related by a b i-sim p licia l homotopy NX . (U .)® (A [0 ]x A [ l]) -» Nx _ (V .), where Q*X t =

s o th at Nx _ (f )* = Nx > (g ) * .

By definition, an e x a c t seq u en ce of ab elian p re-sh eaves P3

0 on E t(X .)

P3(U) -» 0

is a seq u en ce with the property that

is e x a c t for every

U -> X n in E t ( X .) .

0 - P l - P2 -

0 -> P j ^ ) -> P2 (U)

T hu s, such a short e x a c t

seq uen ce induces a short e x a c t seq u en ce of bi-com plexes

0 -> P j(N x (U .)) ->

P2(N x (U .)) ^ P 3(N x (U .)) -> 0 for any U. -> X ; this short e x a c t seq uen ce yields a long e x a c t seq uen ce

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

22

ftJ.))) — • By e x a c tn e s s of colim as d iscu sse d ab ove, we conclude the long e x a c t U.^X. seq u en ce ••• -* H ^ X . ,P j ) In other w ords,

H*(X. ,P 2> - H ^X . ,P 3) - Hi+1(X . .P ^ -» ••• .

V* H (X . , ) is a 5-fun ctor. ■

As we will s e e in the next proposition, there is a s p e ctra l seq u en ce analogous to th at of Prop osition 2 .4 which re la te s C ech cohomology of X . with that of each of the X PROPOSITION 3 .2 . p re s h e a f on X . .

L et X.

, n > 0. b e a sim p licia l s c h e m e and P

an a b elia n

T h en there e x is t s a first quadrant s p ec tra l s e q u e n c e

E f ' 1 = i W x ^ P g ) = > Hs + t(X . ,P ) w here Pg

is the restrictio n of P

P ro o f. F o r any covering to the bicom plex

to E t(X g ) .

U. -> X . , there is a sp e ctra l seq u en ce a s s o cia te d

P (N X (U .)) of the form

E ^ O J . ) = Ht(Pg(N x (Us ))) = > Hs + t(P (N x / U . » ) •

B e c a u s e two maps U. 3 V.

over X .

induce maps

P (N X (V .)) 3 P (N X (U .))

which are related by a filtration -p reservin g homotopy, we conclude that two su ch maps induce the sam e map

{E ^ C V O U

-

.

T h erefore, we may take the colim it of the s p e ctra l seq u en ce indexed by the left directed categ o ry of co v erin g s

U. -> X .

maps to obtain the sp e c tra l seq u en ce

and eq uivalen ce c l a s s e s of

23

3. COHOMOLOGY VIA H YPERC O V ER IN G S

E ® '1 = colim H^PgCNx (U s ))) = > Hs + t(X . ,P ) .

To conclude the proof of P ro p o sitio n 3 .2 , it su ffice s to verify for any s ,t > 0 that the natural map

colim H ^ C N x (U s ))) - H ^ X g .P g ) = colim H ^ P ^ x

(where the first colim it is indexed by coverin gs by coverin g s

W -> X g ) is an isomorphism.

U. = r * '( W s ) -> X .

and the second

F o r th is, it su ffice s to observe

that if W -> X g is e ta le and s u rje c tiv e , then W ^ X S where

U. -> X .

(W )»

Ug -> X g fa cto rs through

(c f ., Prop. 1 .5 ). ■

The failure of C ech cohomology to equal sh eaf cohomology a ris e s from the fa ct that a family of coverings U ’s

U -» X

becom e “ arbitrarily fin e ” while the

can have the property that the Nx (U )^ ’s

do not becom e arbi­

trarily fine for some k > 1 . F o r exam ple, the fa ct that the

U ’s

a c y c lic need not imply that the U x x U ’s

T his problem

becom e a c y c lic .

become

is circum vented by introducing h ypercoverin gs, the following g en eraliza\/ tion of C ech n erves. DEFINITION 3 .3 .

Let

co v erin g

is a b i-sim p licial schem e over

U .. -» X .

X.

An (e ta le ) h y p er­

be a sim p licial sch em e.

X.

with the property

that

UcO* -> X b is a hypercovering of X b for ea ch s > 0 ( i .e ., U Oj fL -» X X ( c o s k ^ U g )t is e ta le su rje ctiv e for all t > 0 , where c o s k j Ug = X g ). The homotopy categ o ry of hypercoverings of X . , denoted

categ o ry whose o b je cts are hypercoverings

U .. -> X.

eq u ivalen ce c la s s e s of maps of hypercoverings of X .

H R (X .),

is the

and w hose maps are ( i .e ., of b i-sim p licial

sch em es over X . ) where the eq uivalen ce relation is generated by pairs of maps U .. =» V ..

related by a sim p licial homotopy

U .. ® (A [0] x A [1 ]) -> V ..

over X . . ■ We re ca ll that the homotopy categ o ry of C ech nerves of coverings U. -» X .

is a left d irected cate g o ry .

A g en eralization of left directed

24

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

categ ory is that of a left filterin g ca teg o ry . A categ o ry

C is left filtering

if (i) for every pair of o b je cts

c " in C together

c , c ' in C ,

there e x is ts

with maps c «- c"-> c ' ; and (ii) for every pair of maps c ' 3 c e x is ts -> c

c " ^ > c ' in

C such that the co m p osites

in C , there

are equal ( c "

c " - > c '= > c

is called a left e q u a liz e r of c ' 3 c ). In the proof of Proposition 3 .2 we used the fa c t that every covering

W -» X g is dominated by Ug -> X g where

U. = r^ * (W ).

We g en eralize this

relation sh ip betw een the left d irected ca te g o rie s of coverings of X . coverin gs of X g a s follow s.

and

A functor F : C -> D is said to be left final

provided th at (i) for every o b ject d of D , there e x is ts an o b ject c C and a map F ( c ) every pair of maps

d in D ; and (ii) for every ob ject c F (c )3 d

in D ,

there e x is ts

that F (y ) is a left eq u alizer of the given maps

in

in C and

y:c'->c

in C such

F (c ) 3 d .

The u sefu ln ess of th e se g en eralizatio n s is that if C is left filtering, then the colim it indexed by the opposite categ o ry of C ,

colim , is an

c° e x a c t functor on the categ o ry of ab elian group valued functors on C ° ; m oreover, if

C and D are left filterin g,

F : C -> D is left final, and

P : D ° -> Ab is any functor, then the natural map colim P ° F an isomorphism (c f ., [8], A .1 .8 ).

c 0

colim P

is

^0

P rop osition 3 .4 re v e a ls the relev an ce of this d iscu ssio n to the categ o ry H R (X .) of Definition 3 .3 . PROPOSITION 3 .4 .

L et X.

be a sim p licia l s c h e m e .

left filterin g and the restrictio n map H R (X .) -> H R (X n)

T h en H R (X .)

is

is left fin a l for

ea ch n > 0 . P roof.

If U .. -> X .

and V .. -» X .

(defined by (U .. x x V ..) g t = U .. -» X .

are h ypercoverin gs, then t x X ^s t

and V .. -> X . . If U .. 3 V ..

U ..x -^ V ..-> X .

a hypercovering mapping to

are two maps of hypercoverings of

X . , we use the con stru ctio n of ([7 ], V .7 .3 .7 ) to co n stru ct the left eq u alizer. Namely, for ea ch Ws

s > 0,

there is a naturally defined sim p licial schem e

= H om (X s ® A [l],V g ) over X g with the property that

3. COHOMOLOGY VIA H YPERC O V ER IN G S

25

H om (Z .,H om (X s ® A [l],V SB)) - Hom (Z.® A [1],V S T.

is s p e c ia l over Y

if Z n maps su r-

je ctiv e ly onto the fibre product of ( c o s k ^ ^ Z . ^ -» ( c o s k ^ j T . ) 0 ), a s seen for exam ple in [29], 1 .3 .

product IK. of W.. -» V . . x x V .. U ..

is a left eq ualizer of U .. 3 V .. .

T o prove that H R (X .) -» H R (X n) is left final, we use P rop osition 1 .5 to conclude that

C ) sending

W. to

r^ * (W .) (defined by

(r n - ( W-))s ,t = ( r n '( Wt)> s) is right adjoint to the re strictio n functor from b isim p licial sch em es over X . if W.

to sim p licia l sch em es over X

is a hypercovering of X n ,

a hypercovering of X .

(b e ca u se

of pull-backs of W. -> X n ).

. M oreover,

then r ^ ‘(W.)is readily ch eck ed to be (r^ * (W .))

is the

fiber product over X g

T hu s, the left finality of H R (X .) -> H R (X n)

follow s from the following lemma (w hose proof we le a v e a s an e x e rc is e ). ■ LEMMA 3 .5 .

L e t F : C -» D b e a functor inducing a functor HoF : HoC ->

HoD w here maps in HoC a nd HoD a re e q u iv a le n c e c la s s e s of maps of C and D r e s p e c t iv e ly .

If HoC

adm its a right adjoint, then HoF

a nd HoD a re left

filterin g and if F

is left fin a l, m

T he next proposition in d icates that a hypercovering of a schem e should be viewed as a resolution of that sch em e. PROPOSITION 3 .6 . L et

Zjj

U. -> X

L et X

b e a s c h e m e and U. -> X

a hy p er co v erin g .

b e the co m p lex of a b elia n s h e a v e s in AbSh(X) d eterm in ed by * w hose m-th term is the a b elia n s h e a f re p r e s e n te d by Um -» X in

26

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

T h en the natural augm entation map Z u

E t(X ).

exp resses

X^

a s a reso lu tio n in AbSh(X)

in d u ced by U Q -» X

-> Z * of Z .

* P roof. T o prove that show that x * ( Z u

X^

-» Z -» 0 is e x a c t in A b S h (X ),

it su ffice s to

*

-»Z -»0) in AbSh(Spec k) is e x a c t for every geom etric *

point x :S p e c k - > X x :S p e c k -> X

(by P rop osition 2 .2 ).

and let W. = U. x x S p eck -> S p eck

U. -> X by x .

Then x * ( Z u

*

^ Z -> 0 ) eq uals

AbSh(Spec k ) . B e c a u s e E t(S p ec k) has o b ject, the global s e ctio n functor Y e t,

F ix a geom etric point denote the pull-back of

Zw *

prove that

AbSh(Spec k)->Ab

W. (ignoring the sch em e stru ctu re).

C ^ W .) -> Z -» 0 is e x a c t.

0 in

id : Spec k -»Spec k a s an in itial

Z w (Spec k) is the free ab elian chain com plex *

sim p licial s e t

Z

is an eq u iv alen ce. C ^ W .) on the T hus, it su ffice s to

T his follow s from the ob servation

that W. is the co n tra ctib le Kan com plex b e ca u s e it s a tis fie s the hyper­ covering condition that

^ (c o s k ^ W .)^

be su rje ctiv e (a s a map of s e t s )

for a ll t > 0 . ■ Using P rop osition 3 .6 , we prove that a sim p licial schem e and its h ypercoverings have the sam e cohom ology. PROPOSITION 3 .7 . hyper co v erin g. AbSh(X .)

L et X.

be a s im p licia l s c h e m e and U .. -> X .

T h en A U .. -> X .

be a

in d u ces an isom orphism of 8-fu n cto rs on

* _ H (X . , )

* H (U .. ,

) .

P roof. If F e A bSh(X .) , then H °(X . ,F ) = K e r (F (X Q) 5 F ^ ) ) , w hereas H °(U .. ,F ) = K e r(F (U 0>0) 5 F ( U l f l )) = K e r (d * -d * : F ( U 0 0) - F ( U l f l ) ) . B ecau se

U n -> X n is a hypercovering for e a ch

n > 0,

the sh eaf axiom

im plies th at K er(F (U n>0) 5 F (U n>1)) = K e r(F (U n>0) 5 F (U n>0x X n Un>0)) equals

F ( X n) .

C onsequently,

H °(U .. ,F ) = K e r(F (U 0>0) - F ( U 1 (1 )) - X K e r (F (X 0) 5 F ( X j ) ) = H °(X . , F ) .

3. COHOMOLOGY VIA H YPERC O V ER IN G S

27

Since the restrictio n functor A bSh(X .) -> A bSh(U ..) is e x a c t, is a 5-functor on A b S h (X .).

H *(U .. , )

T hus, it su ffice s to prove that H *(U .. ,1) =

H *(X . ,1) for I in je ctiv e in A b S h (X .).

B ecause

Us

is e ta le , the

restrictio n functor has an e x a c t left adjoint y : Ab(Us

-> A b (X g ) defined

by sending a sh eaf G on Ug t to the sh eaf a s s o c ia te d to the presheaf (V -»Xg ) k ©G(V ^ U g t ) , where the sum is indexed by the s e t of maps V -> Ug j. over X g . C onsequently, the re strictio n to Ug t of an in je ctiv e on X s

is in je ctiv e .

T herefore,

P rop osition s 2 .4 and 2 .5 .

H * (U .. ,1) = H *(A U .. ,1) - H *(I(A U ..)) by

To compute

H *(I(A U ..)) = H * (I(U ..)),

we

employ the sp e ctra l seq u en ce E f ’4 = H ^ C U ^ ) ) = > Hs + t(I(U ..)) . B ecau se

Ig is in jectiv e in AbSh(Xg ) and b e ca u s e

H om ^g^^

sjc>IS)»

Is (Ug ) =

Prop osition 3 .6 im plies that H*(IS(U .)) = 0 for

t > 0 and H °(Ig(U g )) = Is (X g ) for e ach H *(I(U ..)) = H *(I(X .)) which equals

s > 0.

C onsequently,

H *(X . ,1) by the remark following

P rop osition 2 .4 . ■ We now prove that sh eaf cohomology can be computed using hyper­ co v erin g s.

F o r sch e m e s, this theorem was first proved by J . - L . Verdier

in [7], V .7 .4 .1 . THEOREM 3 .8 .

L et X.

b e a s im p licia l s c h e m e .

T h en there is a natural

isom orphism of 8 -functors on A bSh(X.) H * (X .,

)

w here the colim it is in d ex e d by U ..

colim H *( (U ..)) in H R (X .).

P roof. A s seen in the proof of P rop osition 3 .7 , for any F e A bSh(X.) and any V .. in H R (X .), H °(X . , F ) - = .H ° ( V ..f F ) = H ° ( F (V ..)) » colim H ° (F (U ..)); m oreover, for any in jectiv e

I e A bSh(X .) and any

V ..

in H R (X .),

H * (X ., I)-==»H*(V.., I ) ^ H * ( I ( V . . ) ) ^ c o l i m H *(I(U ..)). T hus, it su ffice s

28

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

to prove that a short e x a c t se q u en ce

0 -»

-» F 2 -» F 3 -» 0 in A bSh(X.)

determ ines a long e x a c t seq u en ce ••• Let

colim H ^ F ^ U ..) ) -» colim H *(F 2 (U ..)) -> colim H *(F 3(U ..))

Gj = cokerC Fj->F2) a s p resh eav es on E t(X .) , and let G3 =

coke^G-^ ->F3) ,

so

that 0 ->

of p resh eaves on E t ( X .) .

-> F 3 -» G3 -» 0 is a short e x a c t seq u en ce

F o r each

U. . ,

0 -> F 1(U ..) -» F 2(U ..) -> G i(U ..) -» 0 ,

0 -» GjCU..) -> F 3(U ..) -> G3(U ..) -> 0 are e x a c t, thus yielding long e x a c t seq u en ces in cohom ology.

B ecau se

H R (X .) is left filtering and b e ca u se

th ese long e x a c t seq u en ces are functorial on H R (X .),

we conclude that it

suffices to prove that colim H *(G 3(U ..)) = 0 . L e t a e G3(U n n) rep resen t a given cohomology c la s s Hn(G 3( U ..) ) ,

some

n>0,

and U ..

a in

in H R (X .). B e c a u s e the sh eaf

a s s o c ia te d to the p resheaf G 3 is the zero sh e a f, we may ch o o se an e tale su rje ctiv e map W -> Un n su ch that a claim that tion 1 .7 ).

T

Un

n‘ (W) = V. -> Un

re s tric ts to

0 in G3(W ). We

is a s p e c ia l map over X n (s e e P ro p o si­

Nam ely, the appropriate su rje ctiv ity in dimension t > 0 is

equivalent to the lifting of geom etric points, which is equivalent to lifting maps from S p e ck ® A [ t ] . T his can be readily ch ecked using the definition of r ^ * ( C onsequently,

) and the ad jo in tn ess of cosk^ ^ ( ) and s kt l ( ) .

U '. = r ^ * (V .)

th at the fibre product U7. of X .

r * * ( U n ) is a s p e c ia l map over X . , so

of U.'. -» r ^ * ( U n ) 0.

T h en there is a

natural isom orphism of 8 -functors on A bSh(X .) H *(X . , ) ^ P roof. ings

H *(X . , ) .

The in clu sion of the homotopy ca te g o ry of C ech nerves of co v e r-

V. -> X .

U .. ^ X .

into the homotopy categ o ry

H R (X .) of hypercoverings

induces a natural transform ation H * (X .,

colim H *( (U ..)) = H * (X ., F e A b S h (X .),

) = colim H *( (Nx (V.)))

) of 5-fu n ctors on A b S h (X .).

F o r any

th is map is induced by a map of filtered co m p lexes, and

thus may be identified with the map on abutments of a map of sp e ctra l seq u en ce F y g .F\g ) = > Hs + t(X . ,F ) E .M f ’1 = = h ifV yX

E f * 1 = H ^ X S’ g .FS ) = > Hs + t(X . ,F ) where the first sp e c tra l seq u en ce is that of Prop osition 3 .2 and the 'E j- te r m is identified using the isomorphism

H *(X g ,F s )

colim H *(F g (Us )) provided by the left finality of H R (X .) -> H R (X g) . The corollary now follow s by applying A rtin ’s theorem to conclude that this map of sp e ctra l seq u en ces is an isomorphism. ■ F o r the sak e of co m p leten ess, we determine the e ffe ct of colim H*( (U ..)) on ab elian p resh eav es with the aid of Theorem 3 .8 . C O R O L L A R Y 3 .1 0 . (ab. g rp s.) n P (U j) ifl

L et X.

be a s im p licia l s c h e m e , and let P : E t ( X .) ° h

b e an a b elia n p re s h e a f with the property that P ( II U -) = lei

for any

II Uj -> X n in E t ( X . ) . iel

d en o te the s h e a f a s s o c ia t e d to P natural isom orphism

L et

P # : E t ( X .) ° - (ab. g rp s .)

(c f. [5 9 ], II.2 .1 1 ).

T h en there is a

30

E T A L E HOMOTOPY O F SIM P LIC IA L SCHEMES

H *(X . , P #) — » colim H * (P (U ..)) w here the colim it is in d e x ed by U .. -» X .

in H R (X .) .

P ro o f. By Theorem 3 .8 , it su ffice s to prove that the natural map P -> P # determ ines an isomorphism of ch ain com p lexes colim P (U ..) -> colim P #(U ..) . B ecau se

P

and P # have isom orphic s ta lk s at every geom etric point and

b e ca u se

P

“ com m utes” with d isjoin t unions, we conclude the following:

If a 6 P (U S t ) goes to 0 £ P #(U g t ) , then there e x is ts an e ta le s u rje ctiv e map U '-> U g j. su ch that a re s tric ts to

0 £P(U0;

and if /3 £ P #(Ug

then there e x is ts an e ta le su rje ctiv e map U"-> U tion of (3 in P #(U ")

f su ch that the re s tric S, I is in the image of P ( U " ) . As argued in the proof

of Theorem 3 .8 , given any e ta le s u rje ctiv e map W -> U S, fL there e x is ts a map of hypercoverings

U.'. ^ U . .

W -> Ug t . C onsequently, phism for any s ,t > 0

,

such that Ug ^

Ug ^ fa cto rs through

colim P (U g t ) -> colim P #(U g p

as required. ■

is an isom or­

4.

E T A L E T OPOLOGICA L T Y P E

As we observed in the la s t ch ap ter, the sh eaf cohomology of a sim p licial sch em e

X.

is determined by its h ypercoverings.

top olog ical type of X . , ( X .) e ^,

The e ta le

is e s s e n tia lly the in verse sy stem of

sim p licial s e ts given by applying the con nected component functor to the h ypercoverin gs.

As we s e e in Chapter 5, the sh eaf cohomology of X.

with lo cally co n stan t c o e fficie n ts can be computed from the homotopy type of ( X .)e t .

\/ Our use of general hypercoverings rather than only C ech nerves does

com p licate our co n stru ctio n , although this com plication is primarily one of notation.

The reader is urged to con sid er only C ech nerves when in itially

con sid erin g the e ta le to p o lo g ical type:

the in verse system of sim p licial

s e ts a s s o c ia te d to C ech nerves (the C ech to p o lo g ica l type) is shown in P rop osition 8 .2 to be weakly equivalent to ( X .) et in most c a s e s of in terest.

On the other hand, there are s e v e ra l important ad vantages of our

definition of the e ta le to p ological type using general h ypercoverings. Namely,

( X .) e ^ has the “ c o r r e c t” weak homotopy type for all lo cally

noetherian sim p licial sch em es and ( )et

is indeed a refinement of Artin-

M azur’s e ta le homotopy type (P ro p o sitio n 4 .5 ).

P erh aps most important,

our con stru ction applies to other s ite s (e .g ., the Z arisk i s ite ) in which \/ C ech cohomology differs from derived functor cohom ology. T his co n stru c\/ tion of the e ta le to p o lo g ical type is based on that of the C ech to p o lo g ical type first appearing in [34] and [51].

The e s s e n tia l idea, su ggested by

work of S. Lubkin [5 4 ], is to “ rig id ify ” cov erin g s by providing e ach co n ­ nected component with a distinguished geom etric point. B e c a u s e the e ta le to p o lo g ical type is not a sin g le sim p licial s e t (or s p a c e ), e ta le homotopy theory is somewhat off-putting at first g lan ce. 31

As

32

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

defined in Definition 4 .4 ,

( X .) e t is a pro-sim p licial s e t (a functor from a

left filtering categ o ry to sim p licial s e ts ) .

Although pro-ob jects were first

sy ste m a tica lly employed in e ta le homotopy (c f. [8], Appendix), the s p e c ia l c a s e of pro-groups has been widely used in g alo is cohomology [66],

At

first acq u ain tan ce with su ch p ro -o b jects, one is tempted to take an in­ v erse lim it.

Not only does one lo se stru ctu re once the in verse limit is

applied, but a lso one obtains in co rre ct in v arian ts:

the cohomology of a

p ro-sim p licial s e t is a kind of continuous cohomology of its in verse lim it. The know ledgeable reader will reco g n ize the c lo s e analogy of our c o n ­ stru ctio n of ( X .) e j. to shape theory in topology [28]. sid ers a “ n ic e ” to p o lo g ical s p a c e

T

In fa c t, if one co n ­

(e .g ., a finite C.W. com plex) and a

“ h ypercoverin g” of T with each component a co n tractib le open su b se t of T ,

then the sim p licia l s e t of con n ected com ponents of this hyper­

covering has the “ s a m e ” homotopy type a s

X

([8 ], 1 2 .1 ).

We begin with the following proposition which provides the rigidity of our con stru ction .

The to p o lo g ical analogue of P rop osition 4 .1 is the fa ct

that a map of con nected covering s p a c e s

X " of a s p a c e

X

is

determined by its value on a sin g le point. PROPOSITION 4 .1 . and V -» X

be a sc h e m e ,

eta le a n d sep a ra ted .

equ a l if f °u = g °u P roof.

L et X

U-^X

eta le with U c o n n e c te d ,

T h en two maps f,g : U -» V

g a s s e c tio n s of the projection

Such se c tio n s are open ([5 9 ], 1 .3 .1 2 ) and clo se d (b ecau se B ecau se

a re

for som e g eo m etric point u : Speck -» U .

We interpret f and

sep arated ).

over X

V

U x V -> U . X X

is

U is co n n ected , we thus may identify a se ctio n

with a ch o ice of con nected com ponents of U x V isom orphic to U . X B ecau se

u and f ° u

(re sp e ctiv e ly ,

g ° v ) determ ine a geom etric point of

the connected component of U x V corresponding to f (re s p e c tiv e ly , X f equals

g whenever f ° u

equals

g°v. ■

g ),

4. E T A L E TO PO LO G IC A L T Y P E

33

The reader should bew are th at P ro p o sitio n 4 .1 is fa ls e if one merely assu m es that f and take

g agree on a sch em e-th eo retic point.

X = Spec k and U = V = Spec K with

K /k

F o r exam ple,

a finite g alo is exten sio n ;

then each d istin ct automorphism of K over k determ ines a d istin ct map from U to

V , but th e se d istin ct maps of co u rse ag ree on the unique

schem e th eo retic point of U . We next introduce the definition of a rigid (e ta le ) covering of a schem e X . B e c a u s e we sh all employ the uniqueness property of Prop osition 4 .1 , we index the con n ected com ponents by the geom etric points of X and provide each con nected component with a distingu ish ed geom etric point. A first problem a ris e s in that an e ta le open U of X joint union of its co n nected com ponents.

may not be a d is ­

A secon d problem is the

aw kw ardness th at the co lle c tio n of all geom etric points of X We re ca ll th at a sch em e X

is not a s e t.

is said to be lo ca lly n o etherian provided

that it is a union of affine noetherian sch em es ( i .e ., sp e ctra of noetherian rings).

B e c a u s e the Z a risk i to p o lo g ical s p a ce of a lo cally noetherian

schem e has the property th at every point has a sy stem of neighborhoods in which each d escen ding chain of clo se d s u b s e ts is fin ite, we readily conclude th at each sch em e-th eo retic point of a lo ca lly noetherian schem e has su ch a neighborhood which is a ls o irreducible. th at the con n ected com ponents of X

T his e a sily im plies

are open as w ell as clo se d ([4 7 ],

I. 6 .1 .9 ) . To avoid s e t th eo retic problems, we ch o o se for e a ch c h a ra c te ris tic p > 0 an alg e b ra ica lly clo se d field

12p which is su fficien tly large to

con tain su bfields isom orphic to the residu e fields of c h a ra c te ris tic every sch em e we co n sid er.

p of

(At this point, s e t th e o rists would su g g e st we

fix a “ u n iv e rse ” and con sid er only th ose sim p licial sch em es which in each dimension lie in th is u n iv e rse .) Once su ch a ch o ice is made, we define the s e t of g eo m etric points of X , denoted geom etric points

x : Spec 12 -> X

X , to c o n s is t of all

where 12 = 12p with

p the residue

c h a ra c te ris tic of the image schem e th eo retic point of X .

34

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

With th e se p relim in aries, we introduce rigid co v erin g s. DEFINITION 4 .2 . scheme

X

A rig id co v erin g a : U -> X

of a locally noetherian

is a disjoint union of pointed e ta le , separated maps ax : Ux ' ux

where each

Ux

Vx € X

X ’x '

is co n nected and ax °u x = x : Spec 12 -»X . A map of

rig id co v erin g s over a map f : X -> Y

of sch e m e s,

cf> : (a : U -»X) ->

(j8 : V -»Y) is a map cf): U -» V over f su ch that 0 ° ux = vf(x ) f ° r x e X . If U -> X and

V -> X are rigid coverings of X

and Y

over a

R

third sch em e

S , then the rig id product U x V - ^ X x Y S

the open and clo se d immersion of U x V ^ X x Y S

union indexed by geom etric points

is defined to be

S

given a s the d isjoint

S

x x y of X x Y

of

S

ax x

where

(U

:

(ux

x Vy)0 , ux x vy

X x Y , xxy

x V ) 0 is the con n ected component of Ux x V

s y

s y

containing

ux x vy • ■ Of co u rse, P rop osition 4.1 implies th at th ere is at most one map between rigid co v e rin g s.

T his im plies that R C (X .),

rig id co v erin g s of a sim p licial sch em e

the ca tego ry of

X . , is a left directed categ o ry .

We proceed to define rigid hypercoverings in su ch a way that they too determ ine a left directed categ o ry .' PROPOSITION 4 .3 .

L et X.

b e a lo ca lly n o eth eria n sim p licia l s c h em e .

A rig id h y p erco v erin g U .. -» X .

is a h y p erco v erin g with the property that

Us ,t - (c o s k j® Us )t is a rig id co v erin g for e a ch s ,t > 0 s u c h that any map a : A [s '] -> A [s]

4. E T A L E TO PO LO G IC A L T Y P E

35

in d u ces a map of rig id c o v erin g s over a : (cosk^._® Ug for ea ch t > 0 .

A map of rig id hy p er c o v erin g s over a map f : X . -» Y .

a map of b isim p licia l s c h e m e s 0s t :U st-V s t

cf): U .. -> V ..

is

over f su c h that

X is a map of rig id c o v e rin g s over c o s k t_ 10 : ( c o s k ^ U g )t

Y -» (c o s k ^ jV g )^. for ea ch s ,t > 0 . of X . , H R R (X .),

T h e ca teg o ry of rig id hy p er c o v erin g s

is & left d ire c te d ca tego ry .

P roof. B e c a u s e any sim p licial map cf>s : Ug the

-> ( c o s k ^ Ug ')j.

(t-l)-tru n c a tio n of

determ ines

Vg

has the property that

co sk t l 0 s

, we conclude by

Prop osition 4 .1 th at there is at most one map betw een any two rigid hyper­ coverings over a given map of sim p licial sch e m e s.

To prove H R R (X .)

left d irected , it thus su ffice s to observe that if U .. -> X .

is

and V .. -> X .

R are rigid hypercoverings then their rig id product U .. x V .. -> X . is a rigid X. R R hypercovering mapping to both, where (U .. x V ..)g t -» (co sk t l (Us x Vs )T X. ’ • Xs is defined to be the re strictio n to

(c o s k t l (Ug

R x Vg ))t C ‘ xs

(c o s k t l Us )t x (co sk t l Vs )t of the rigid product over X g of xs Us ,t - (c o s k t - l Us .) t

and Vs ,t - (c o s k t - i Vs . ) f ■

We remind the reader that a p ro -ob ject of a categ o ry from some sm all left filtering category to p ro-object F : I -» C by {F- ; i e l i

of p ro -ob jects from F : I -> C to lim colim Homc ( F ( i), G (j) ) ; J

C . We sh a ll often denote a

or simply

ignore the s e t th eo retic requirem ent that

C is a functor

iF^i,

and we sh all usually

I be a sm all categ o ry .

A map

G : J -> C is an elem ent in the s e t

in other words, a map from F

to G is a

I

com patible c o lle c tio n of elem ents (indexed by J ) in colim Homc ( F ( i ) , G ( j)), each of which is determined by a map

36

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

F (i ) -> G(j) for some from F : I

C to

A strict map of p ro -o b jects

i (depending on j e J ).

C is a pair co n sistin g of a functor a : J -» I

G:J

: F °a -> G . Of co u rse , a s tr ic t map of

and a natural transform ation

p ro-ob jects determ ines a map of p ro -o b jects. If F : I -> C is a pro-object and if a : J -» I is a functor between left filtering c a te g o r ie s , then (a : J -* I , id : F °a -> F ° a ) F

to F o g .

is a s tr ic t map from

If the functor a : J -» I is left final, then we readily con clu d e

the e x is te n c e of a (not n e c e s s a rily s tr ic t) map of p ro-ob jects inverse to th is ca n o n ica l map.

F °a -> F

In other w ords, the s tr ic t map F -» F °a

is

an isomorphism of p ro -o b jects, even though it may not have a s tr ic t in verse. \/ * The C e c h top o logica l type of X . is defined to be the p ro-sim plicial s e t A (X .)ret = 77ANx ing a rigid covering

: R C (X .) -> ( s . s e t s ) (notation is that of [34]) sen d ­

U .. -» X .

to the sim p licial s e t

77(A(NX (U .)))

in dimension t a s the s e t of con n ected com ponents of the

given

t-fold fibre

product of Uj. over X j.. T his definition su g g e s ts our definition of the e ta le to p ological type. DEFINITION 4 .4 .

Let

X.

be a locally noetherian sim plicial scheme.

The eta le topologica l type of X . sim p licial s e t

is defined to be the following pro-

( X .) et - 77° A : H R R (X .)

sending a hypercovering U ..

of X .

(s . s e ts )

to the sim p licial s e t

con nected components of the diagonal of U .. s e t of co n n ected com ponents of Un

).

(s o that

If f : X . -> Y .

7t(AU..) of

77(AU..)n is the is a map of lo ca lly

noetherian sim p licial sch e m e s, then the stric t e ta le to p o logica l type of f is the s tr ic t map

fe t : ( X . ) e t , ( Y . ) et

given by the functor f* : H R R (Y .) -» H R R (X .) and the natural transform ation ( X .) e t ° f * -> ( Y .) et induced by the natural maps in H R R (Y .).

f * (V ..) -> V ..

The e ta le top o lo g ical type functor ( ) e j .:( l o c . noeth. s . sch em es) -> (p ro -s. s e ts )

for V .. -> Y .

4. E T A L E T O PO LO G IC A L T Y P E

sen ds a lo cally noetherian sim p licial schem e f : X . -> Y .

X.

37

to ( X .) e {- and a map

to the map of pro-sim p licial s e ts determined by f ^ . ■

In the above definition, the functor f* : H R R (Y .) ^ H R R (X .),

the

rig id pu ll-ba ck functor, is defined in term s of the pull-backs of rigid co v erin g s.

If V -> Y

f*(V -»Y) = U -> X

is a rigid covering and f : X -> Y

is the disjoint union of pointed maps

(Vf(X ) x X ) 0 , f(x) x x ^ X ,x component of Vf. . x X Y V.. -♦ Y.

is a map, then

where

(V £ ^ x X ) Q is the con nected

con tain in g the geom etric point f(x) x x . If

is a hypercovering of Y. , then f*(V .. -^Y.) = U .. -> X .

is

defined for any s ,t > 0 by Us ,t - ( c o s k ^ U s A If X .

= f*(VSft - ( c o s k ^ V g .) ) •

is a sim p licial sch em e provided with a ch o sen geom etric point

x : Spec Q

then

X.

(or, more p re cise ly

po in ted sim p licia l s c h e m e .

C learly , if X .

(X . , x ) ) is said to be a

is a pointed lo cally neotherian

sim p licial sch em e, then ( X .) e £ is naturally a p ro-object in the categ o ry (s . s e ts ^ ) of pointed sim p licial s e t s .

M oreover, if f : X . -> Y .

is a

pointed map of pointed lo cally noetherian sim p licial sch e m e s, then f ^ is naturally a s tr ic t map of p ro-(s. s e t s ^ ) . O riginally, the e ta le homotopy type of a sch em e X ht = 7 7 °A :H R (X ) - » H,

X was defined to be

where H is the homotopy categ o ry of sim p licial

s e ts defined by inverting weak e q u iv alen ces ([8 ], 9 .6 ). PROPOSITION 4 .5 .

L et X

(X ® A [0 ])e £ e pro-(s. s e ts ) X ^ € Pro_W

be a lo ca lly n o eth eria n s c h e m e , let

be d e fin e d a s in D efin itio n 4 .4 , an d let

d e fin e d a s a b o v e . If (X ® A [0 ])e £ is v iew ed in pro-K

applying the fo rg etfu l functor (s . s e t s ) - >K, ph ic to X ^

by

then (XA[0])e ^ is isom or­

in pro-K .

P ro o f. We readily verify th at a rigid hypercovering U .. -* X ® A [0 ] is of the form (U. -> X )® A [0 ],

where U. -» X

is a rigid hypercovering of X

38

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

(i .e .,

Un -> (co sk n l U .)n is a rigid coverin g for n > 0 ).

T hus, it

su ffice s to verify that H R R (X) -> H R (X ) is left final, where H R R (X ) is the left d irected categ o ry of rigid hypercoverings of X . If g : U -» V is a s u rje ctiv e map, then a “ c h o ice of right inverse to g* ” is a function g” 1 : V -> U right in verse to g^ : U -> V sending u : Spec 12 -> U to

g *(u ) = u ° g : Spec 12 -> V . If g : U

and s u rje ctiv e , and if g~*

V is e tale

is a ch o ic e of right in verse to g^ , we let

= J J (U y .g ^ C v ) ^ v ,v ) v fV

be a rigid covering of V , open of U containing

where Uy is a co n n ected , sep arated Z a risk i

g ~ * (v ).

C learly ,

R ( g , g ^ ) -►V fa cto rs naturally

through g : U -> V . U sing this co n stru ctio n , we exh ib it for U. -> X in H R R (X ) naturally mapping to U. -> X . Define R (g 0,g o * ) -* ^ , where

g Q: U Q -> X

a ch o ice of right in verse to

g 0* .

in HR(X) , R (U .) -» X R (U .)0 -> X

to be

is the augm entation map and g ^

is

Inductively, define R (U .)n ->

(c o s k n l R (U .))n to be R (g n, g ^ ) -> ( c o s k ^ R C U .) ^

where g n is the

projection of the fibre product of (c o s k n l R (U .))n -» (c o s k n l U .)n «- Un onto

(co sk n l R (U .))n and g“ * is induced by the degeneracy maps on

(s k n_ 1R (U .))n C (c o s k fl_ 1R (U .))n . T o com plete the proof of left fin ality , let sim p licial sch em es with U. h : H. -> U. 3 .4 .

is s p e c ia l, we co n stru ct a map

in H R (X .) such that h ° f : W. -> U.

is in H R R (X) .

f 0 : W0 ^HQ to be R (s,6~1) ^ H q ,

augm entation map and

e~ l

s a tis f ie s

b;+.°E ~ 1(x )

the distinguished geom etric point above

Uq -> X ).

in H R (X ), and let

be the left equalizer con stru cted in the proof of Proposition

We define

ux

in H R R (X ) and V.

Using the fa c t that H. -> U.

f : W. -> H.

U. 3 V. be two maps of

With e " 1 so ch o sen ,

x

where =

e:Hq^X

ux for all x

f

is the

X

(with

of the rigid covering

h Q° f 0 : WQ -» U Q is a map of rigid c o v e r­

ings of X . Inductively, we let g^ : Kn

L n denote the map induced by

39

4. E T A L E TOPOLOGICAL T Y P E

h from the fibre product Kn of Hn -> (c o s k n l H .)n (c o s k n l U .)n ,q = colim H ^(F(V V..

)) = > colim HP+C1 (F (U ..))

p*

u..

)) => colim HP+C1 (F (V ..)) V..

We show th at this map is an isomorphism.

By P ro p o sitio n s 3 .4 and 4 .5 , it

su ffices to prove th at the re strictio n map H R R (X .) -» H R R (X p ) is left final for each R r * - ( W .) ^ X . P settin g

p > 0.

Imitating the proof of P ro p o sitio n 3 .4 , we define

in H R R (X .) a s s o c ia te d to

W. - X n in H RR(X ) by P P

( R r ^ ( W .) ) c to be the rigid product of the rigid pull-backs of P s. W. -» X via the maps X -> X n indexed by A [ s ]_ . Then ( R r ^ ‘(W .)) -> P b P P P P* Xp facto rs through W. -» Xp so that H R R (X .) HRR(Xp) is left final. ■

40

E T A L E HOMOTOPY OF SIM P LIC IA L SCHEMES

The proof of the following proposition makes frequent use of the uniqueness of maps betw een rigid hypercoverings to insure the validity of sim p licial id en titie s.

The u sefu ln ess of th is proposition will become

apparent when we co n sid er function co m p lexes. PROPOSITION 4 . 7 .

L e t X.

be a lo ca lly n o eth eria n sim p licia l sc h e m e

and let n b e a p o sitiv e in teg er. (X .® A [n])e t

T h en the natural strict map

( X .) et x A[n]

in

pro-(s. s e ts )

is an isom orphism with in v ers e pro v ided by a stric t map ( X . ) ^ x A[n] (X . ® A [n])e t . P roof. B e c a u s e the co sk e le to n functor commutes with d isjoint unions, U .. ® A[n] -» X . ® A[n] defined by (U.. A [n]) S j .L = Ub ^I . ®A[n] b is a rigid hypercovering of X .® A [n ] whenever U .. is a rigid hypercovering of X . . Therefo re, the id en tification of ( X ,) e ^.®A[n] with (X . ® A [n])e ^. ° ( ? ® A[n]) determ ines the s tr ic t map (X .® A [n ]) ^ -» ( X .) ^ x A[ n ] .

To prove the

proposition, it su ffice s to exhibit a right adjoint for ( ? ® A [n ]), p : H R R (X.® A[n]) -> H R R (X .),

so that the adjunction transform ation

( ? ® A [ n ] )° p -> id determ ines a stric t map from (X . ® A [n])ej. ° ( ?® A [n]) = ( X .) ej.xA [n ] to (X .® A [n ])e ^. and (? ® A [ n ] ) is left final. For U.'. -> X .

U .. -> X .® A [n ] in H R R (X. ® A[n]) , we define p(U .. - X . ® A[n]) = in H R R (X .) by se ttin g

Uk

equal to the rigid product in

n+k H R R(X^) indexed by the ( k ) non-degenerate sim p lices x A [n])n+k of

**(U £ + n^ X n+k® {a '| ) in H R R (X k) , where

is th at copy of X k+n® A [n]k+n indexed by

< o, o '> e (A[k] X k+n® { a '}

o ' and where Uk+n is the

restrictio n of Uk+n -> (X . ® A [n])k+n to X k+n® {c r 'i.

If a : iO,---,kS

! 0, ---, mS is a n on-decreasing map, we define the com posite Uk. " a * (Uk+n.) by « /oP V : U m. ^ ^ (U ra+n.) ^ ff^ Uk + n .)'

pr o a : ! ^

->

where

< p , p > e (A [m ]x A [n])m+n and a\ l0,---,k+n| -> { 0 , - - -, m+n! are defined as follow s:

define a ' by a'Q ) = a '( j - 1 ) + 1 for j su ch that

a ( j ) = cr(j—1)

4. E T A L E TOPOLOGICAL T Y P E

and a '( j ) =a X j - 1 ) + a ( a ( j ) ) - a ( a ( j - l ) )

if a ( j )

41

^ a (j-l) ;

define

p: l0,***,m+n! -> {(),•••,mi to be the unique su rje ctiv e , non-decreasing function such that a ' ( j)

=

i and

a (j)

( i { i)

whenever there e x is t s a

=

a ( j + l ) . We readily verify that

=

unique su rjectiv e non-decreasing maps fitting

A|n]

fi

j su ch that

and

are the

\l

inthe commutative squares

------ ^ ----- A[k+n] -------- - ----- ►A [k ]

a

id

a

A[n] - ------ ^ ------ A[m+n] ------- £------►A[m]

B e c a u s e th ese squares commute, a*(U ^+n) covering

a ': Um+n -> Uk+n

a :X m ^X^.

Moreover, if

r e s t ric ts to /**(Uj£+n ) : !0,--*,m i -> {0, •**,p i is

another non-decreasing map, we readily ch eck that (/3 ° a ) ' = fact that U.'.

The

is a well-defined bi-simplicial scheme now follows using

rigidity. We briefly sk e tch the proof of the adjoin tness of ?® A [n ] and p . f :(W ..® A [n ] - * X .® A [ n ] ) -* (U ..-> X . ® A[n])

If

is a map in HR R(X.® A[n]) ,

then we obtain (W.. ->X.) -> ( p (U ..)-» X .) in H R R (X .) by defining \ p(U..)k . ^ a * (U k+n.) as the restriction °f a*(Wk ) .

^k+n: \ + n .® ^ a ^

Uk+n.

to

C onversely, if g : (W.. ^ X . ) -» (p (U ..)-+ X .) is a map in H R R ( X .) ,

then we obtain (W.. ® A[n] -» X . ® A[n]) -> (U .. -^X. ® A[n]) whose restriction to Wk ® l s l ^ U ^

is defined by ) ->

Uk , where < o ,o '> ^ is some non-degenerate simplex of (A[k] x A[n])k+n and where 8 : cr*(U^+n )

Ufc

is induced by 8 : Uk+n -> Uk

with

S : {0,***,k| -» { 0 , * “ ,k+ni any strictly increasing map such that and 8 : A[n]k+n -> A[n]k sends

o ' to

o ° 8 = id

e . ■

The following is an immediate corollary of Proposition 4 . 7 . C O R O L L A R Y 4 .8.

L e t f, g : X. -> Y .

b e two maps of lo ca lly noetherian

sim p licia l s c h e m e s re la ted by a sim p licia l homotopy. d eterm in e the sam e map in pro-K. ■

T h en f e ^ and ge ^

5.

HOMOTOPY INVARIANTS

In th is ch ap ter, we co n sid er the homotopy groups and cohomology groups of the e ta le to p o lo g ical type ( X .) e £-

We identify

ttq

and

with their alg eb raic cou nterp arts (p artial resu lts about the higher homotopy groups are to be found in later ch a p te rs).

In p articular, our

study of the fundamental group requires a review of d e sce n t techniques in the co n text of principal G -fibrations.

U sing our id en tification of funda­

mental groups, we a ls o verify th at the cohomology of ( X .) e j. with abelian lo ca l co e fficie n ts is isom orphic to the cohomology of X . cie n ts in the corresponding lo ca lly co n stan t sh eaf.

with co e ffi­

We point out that

th ese homotopy and cohomology groups are invariants of the homotopy type of ( X .) e j_; by Theorem 3 .8 , the top ological type itse lf determ ines in some se n s e the cohom ology groups of X .

with valu es in any abelian

sh eaf. DEFINITION 5 .1 . sch em e.

For

Let

n > 0,

X . ,x

be a lo ca lly noetherian pointed sim p licial

we define the pro-object of pointed s e ts

* n((X . >XW

= nn ° ( x * »x )e t : H R R (X .) -> ( s e t s ) .

T hu s, for n > 1 , 77fl((X . ,x )e t) is a pro-group; for n > 2 , 7rn((X . ,x )e t) is a pro-abelian group. 7t ° A (U ..) for some 5 .8 ).

F o r any

L e t M be an ab elian lo ca l co e fficie n t system on

U. . eHRR( X. )

n>0,

(s e e d iscu ssio n preceding Corollary

we define

Hn(( X .)e t ,M) = colim Hn(7 7 °A (U :.),j*M )

where the colim it is indexed by j : U.'. ->U ..

42

in H R R (X .)/U .. . ■

43

5. HOMOTOPY INVARIANTS

The reason we co n sid er the homotopy pro-groups rather than their in verse lim its is th at too much information is lo st upon applying the in verse limit functor (which is not even e x a c t, unlike the d irect limit functor colim ).

As we se e in the next proposition,

770((X . ,x ))

is actu ally

(isom orphic to) a pointed s e t. PROPOSITION 5 .2 .

L e t X . ,x

sch e m e and let t7q(X .

,x )

b e a lo ca lly n o eth eria n p o in ted sim p licia l

d en o te the p o in ted s e t of c o n n e c te d com ponents

of the p oin ted sim p licia l s e t n(X. , x ) , 77q(X. the po in ted pro-set 77Q((X . ,x )e t)

,x )

= tTq(7t{X. , x ) ) .

is isom orphic in p ro -(sets^ )

T h en

to the

pointed s e t ?7q(X. ,x ) 770((X . ,x )e t) Furtherm ore, X .

ac 77q(X . , x) .

is a d isjo in t union of non-trivial sim p licia l s c h e m e s

co rresp o n d in g to e lem en ts a e 77Q(X . ,x )

with no X ?

X?

e x p r e s s ib le a s a

non-trivial d isjo in t union. P roof. We ignore the b ase point and prove that the natural map 77Q( ( X .) e t)

77q(X.) is an isomorphism of p ro -se ts. F o r th is, it su ffice s to prove that this map induces a b ijectio n Hom(77Q(X .),S ) -> Hom(770( ( X .) e t ,S) = colim Hom(770(77(AU..)),S)

for any s e t

S . To prove th is, it su ffice s to prove that the natural map

H °(X . ,S) = Hom(770(X .),S ) -> Hom(n-0(n (A U ..)),S ) = H °(U .. ,S)

is a b ijection for any s e t

S . Viewing S a s a co n sta n t sh eaf of s e ts , we

may apply the proof of P rop osition 3 .7 to verify this la s t b ijectio n .

The

secon d a sse rtio n of the proposition is verified by in sp ection . ■ G rothendieck’s alg e b ra ic interpretation of fundamental groups is given in terms of coverin g s p a c e s .

T h o se con nected covering s p a c e s which

correspond to normal subgroups of the fundamental group are principal G -fibrations, where

G is the quotient group.

44

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

DEFINITION 5 . 3 . (d is cr e te ) group. sch em es

Let

X.

be a sirnplicial scheme and let

A p r i n c ip a l G - fib r a tio n over X .

f:X .'n > X .

G be a

is a map of sirnplicial

together with a right actio n of G on X.' over

X.

su ch that a .)

T here e x is ts an e ta le , s u rje ctiv e map U ->Xq and an isomorphism

of sch em es

U x Xq-+U®G

over U commuting with the actio n of G

xo (w here G a c ts on U®G b .)

by right m u ltiplication).

F o r each map a : A[n] -» A[m] in A ,

\ r '

O'

v

m

fn

x m ------

Xn

A map of principal G-fibrations over X . , X ." - > X ',

isomorphism of sim p licial sch em es over X . G.

'

n

fm

is ca rte s ia n .

the square

commuting with the actio n of

We denote the categ o ry of principal G-fibrations over X . In the following lemma, we re-in terp ret the above categ o ry

in terms of the categ o ry

is an

n ( X 0, G) plus “ d e sce n t d a t a .”

by Ei(X. ,G) . II(X. ,G)

We intend the

following d iscu ssio n to be a first introduction to the technique of d e sce n t. L E M M A 5 .4 .

L et X.

the ca tego ry II(X. ,G )

b e a sim p licia l sc h e m e and let G be a group. T h en of p rin cip a l G -fibrations over X .

the ca tego ry < II(X q , G) ; d.data > d e fin e d a s fo llo w s.

is eq u iv a len t to

A n o b ject of

< II(X q , G ) ; d .d ata > is a prin cip a l G -fibration over X Q, q Q: X Q ^ X 0 , togeth er with “ d e s c e n t d a ta ” —nam ely, an isom orphism in I1(X 1,G )

sa tisfy in g d* = d^°dQ

(q0 : X q ^ X 0 ; 0 )

to ( r 0 : X g - > X 0 ;^r)

in II(X 2 , G ) .

-» d ^ q ^ A map from

is a map 0 : X q - > X q

sa tisfy in g the condition i/r ° d * 0 = d * (9 ° 0

in I ^ X p G ) .

in II(X 0, G)

45

5. HOMOTOPY INVARIANTS

P roof.

T o a principal G -fibration q : X. ' - > X .

over X . , we a s s o c ia te

the principal G-fibration q 0 : X Q - > X 0 over X q together with the isom or­ phism 0

given a s the com posite

q1

dQ(qQ)

d * (q Q) . C learly ,

q h> < q q, 0 > determ ines a faithful functor from II(X . ,G) to

< n ( X 0,G); d. data > . C on versely, let q Q: X Q ^ X 0 and

: dQ(qQ) -> d * (q Q) determ ine an

elem ent of < I I ( X 0 ,G ); d. data > . We define

q: X. ' - >X.

by settin g

X^

equal to the fibre product of d Q° ••• ° d Q: X n -> X Q and q 0 : X Q ^ X Q and by settin g

qn : X^ -* X n equal to the projection .

Define s - : X ^ _ j -» X^

to be s j; x id a Y 'q ; define d-i : nX ' -> X n— ' i 1 to be ld;x i dY / aq and define

dfl : X^ -> X ^ _ j

for 0 X .

is a well-defined map of sim p licial sch e m e s. More­

over, using th is e x p licit co n stru ctio n ,

we s e e that if (q 0, 0 ) and ( r Q, if/)

are elem ents of < I I ( X q,G ); d. d a t a > ,

then a map 0 : q Q ^ r Qof sch em es

over X q with ifrod*d - d * 0 ° 0 q h> r .

determ ines a map of sim p licial sch em es

In p articu lar, the G -action on q Q determ ines a G -action on q ,

so that q : XT-> X .

is an elem ent of II(X. ,G ).

The sam e argument

implies th at a map (q Q, 0 ) -» (r Q, t/r) in < II(X q ,G ); d. data > determ ines a map q

r

in II(X . ,G ) . ■

F o r Y .. is a b i-sim p licial

sch em e, we interpret the category

II(A Y .. ,G ) in a sim ilar fashion. PROPOSITION 5 .5 . ( d is c r e t e ) group.

L e t Y ..

be a b i-sim p licia l s c h e m e an d G a

T h en ca tego ry II(A Y .. ,G)

d e fin e d a s follow s.

is eq u iv a len t to the ca tego ry

A n o b ject of

is a p rincipa l G -fibration q : Y.'-> Y Q p ro v ided with an isom orphism cj> : dQ(q)

d*(q)

a map (q, cjT) dition

in

II(Y1 ,G ) sa tisfy in g

(r, \jf) is a map 0 : q -» r

°dQ0 = d * 0 ° 0 .

d* = d* : d qY.' -> d*Y." in I^ Y j

is equivalent to a map 77(U0

),u

d * 0 = d ^ °dQ0 .

47

5. HOMOTOPY INVARIANTS

Since Ug

is a hypercovering of X g for s > 0 ,

the categ o ry of

principal G -fibrations over tt(Us ) is equivalent to the categ o ry of principal G -fibrations over ([8 ], 1 0 .7 ).

Xg

whose re strictio n to

Ug Q is trivial

The seco n d a sse rtio n of the proposition is now immediately

implied by Lemma 5 .4 .

To prove the first a s s e rtio n , it su ffice s to re ca ll

that Hom(7r1((X . ,x )e j.),G) = colim Hom(771(77(AU..),u),G ) by definition and that if X .'^ X .

is a principal G-fibration then there e x is ts

the property that X 'Q x U q 0 - >UQ 0 *s xo (if U - ^ X q triv ia liz e s the rigid covering

U .. -» X with

v ia l principal G-fibration

’

X q - » X q , define

U ..

to be the “ rigid n erv e” of

R F q *(U) -» X . ; cf. proof of C orollary 4 .6 ). ■

Prop osition 5 .6 e s ta b lis h e s the following useful fa ct which we s ta te a s a sep arate co ro llary . groups

We im plicitly u se the fa c t that a map of pro­

{ H - ; i e l i ^ l G j ; j e J i which induces a b ijectio n

Hom(lH^S,K) for a ll groups

Hom(lGj S,K)

K is n e ce ssa rily an isomorphism of pro­

groups. C O R O LL A R Y 5 .7 .

L et X. ,x

sim p licia l s c h e m e .

L et

C

be a p o in ted , c o n n e c te d , lo ca lly noetherian

be any left filterin g su b ca teg o ry of H R(X. ,x )

(th e homotopy ca teg o ry of po in ted hyper co v e rin g s of X . , x ) with the property that for any eta le s u rje c tiv e map U -> X Q th ere e x is t s som e U .. -> X .

in C s u c h that U Q Q fa cto rs through U - » Xq.

functo rs

C -> H R (X. ,x ) H q q , where v ® l : Spec 0 -> //o m (A [l],V Q ) Q is the co n stan t homotopy. If T . a functor

is a sim p licial s e t, then a lo ca l c o e ffic ie n t sy stem on T .

is

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

48

L : (A /T .)° assig n in g to each sim plex

(s e t s )

t m e Tm a s e t

A[m] in A an isomorphism

L (tm) and to ea ch

a : A [ n ] -»

L (a ) : L (a (tm)) -» L ( t n) (c f. [42] for a d isc u s -

sio n 'o f A / T . , the ca te g o ry of sim p lices of T . ).

A map of lo ca l co e ffi­

cien t sy stem s is a natural transform ation of fu n ctors. c la s s of lo ca l co e fficie n t sy stem s isom orphic to to S on the pointed, con nected sim p licial s e t

L

An isomorphism

with fibres isomorphic

T . , t naturally corresponds

to an eq uivalen ce c la s s of homomorphisms n 1(T . ,t) - Aut(S) , where the eq u ivalen ce relation is generated by inner automorphisms of A u t(S ). Such an eq u ivalen ce c l a s s of homomorphisms naturally corresponds to an isomorphism c la s s of principal A ut(S)-fibrations on T . . For

{T? ; i e l! e pro-(s. s e ts ) , a lo c a l co e fficie n t system on !tM

eq u ivalen ce c l a s s of lo c a l co e fficie n t sy stem s on some on T ! and h i

on Tl

t ! , where L*

are equivalent if there e x is ts some k mapping to

i and j in I su ch that the restrictio n s of l ! isom orphic.

and

hi

to

are

B e c a u s e the re strictio n s of a lo ca l co e fficie n t sy stem

a sim p licial s e t

T.

is an

via weakly homotopic maps

th is definition a ls o ap plies to categ ory of sim p licial s e ts .

S. ^ T .

StM e p ro -H , where

L

on

are isom orphic,

K is the homotopy

As d iscu sse d ab ove, if !tM e p ro-(s. s e ts ^ c )

is a pro-object of pointed, con nected sim p licial s e t s , then an isomorphism c l a s s of lo ca l co e fficie n t sy stem s on 1t!S with fibres isom orphic to a given s e t

S is equivalent to an eq uivalen ce c l a s s of homomorphisms

ttjCiTf!) = {ttjCTi)S - A u t(S ). With th e se p relim inaries, we give the following corollary of P rop osition 5 .6 . C O R O L L A R Y 5 .8 .

L et X.

s c h e m e , let U .. -> X .

b e a c o n n e c te d , lo ca lly noeth eria n sim p licia l

b e a hyper co v erin g , an d let S b e a s e t .

T h en

th ere is a natural one-to-one c o rre s p o n d en c e b etw een the s e t of isom orphism

5. HOMOTOPY INVARIANTS

c l a s s e s of locally con sta n t s h e a v e s on E t(X .)

49

with sta lk s isom orphic to

S w hose restrictio n s to Uq q are constant an d the s e t of isom orphism c la s s e s of lo ca l c o e ffic ie n t sy s tem s on ^ (A U ..)

with fib re s isom orphic to

S . In particular, an isom orphism c la s s of lo ca lly con sta n t s h e a v e s on E t(X .)

is in natural one-to-one c o rre s p o n d en ce with an isom orphism c la s s

of loca l c o e ffic ie n t sy s te m s on ( X .) e j- w hich is in natural one-to-one c o rre s p o n d e n c e with an e q u iv a le n c e c la s s of homom orphisms 771((X . ,x ) e f-) -» Aut(S) for any geo m etric point x

of X Q,

w here S is isom orphic to

the s ta lk s . Proof.

A s seen ab ove, a lo ca l co e fficie n t sy stem

sta lk s isom orphic to

By P rop osition 5 .6 , this is naturally equivalent

to a principal Aut(S)-fibration over X .

w hose re strictio n to Uq q is

Such a principal Aut(S)-fibration is e a sily seen to be equivalent

to a map of sim p licial sch em es X Q x U Q0 - U 0 0 ®S. x o

on 77(AU..) with

S is naturally eq uivalent to a principal Aut(S)-

fibration over 7t(A U ..).

triv ial.

L

X .'-* X .

sa tisfy in g 5 .3 .b .) su ch that

Such a map X.'-* X .

is equivalent to the lo cally

’

co n stan t sh eaf of s e ts on E t(X .) with sta lk s isom orphic to sending

U -» X m to the s e t of maps from U to

X^

S defined by

over X m . T his

proves the first a s se rte d eq uivalen ce of c a te g o rie s ; the secon d follow s from th is. ■ U sing C orollary 5 .8 , we now identify the cohomology of ( X .) e |- with co e fficie n ts in an abelian lo ca l co e fficie n t sy stem . P R O P O S I T I O N 5 .9 .

L et X.

b e a c o n n e c te d , lo ca lly noeth eria n sim p licia l

s c h e m e , let M b e a lo ca lly co nstant a b elia n s h e a f on E t(X .) , a nd let M a ls o d en o te the co rresp o n d in g a b elia n lo ca l c o e ffic ie n t sy stem on ( X .) e {_. T h en there is a natural isom orphism H *(X . ,M) ~ P roof.

If L

H * ((X .)e t ,M) .

is an ab elian lo ca l co e fficie n t sy stem on a sim p licial s e t S. ,

then H*(S. ,L )

is defined to be the cohomology of the com plex

50

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

in h

II s

n

fS

L (s ) ! .

If U .. -» X .

is a hypercovering of X .

su ch that the

n

lo cally co n stan t abelian shea^f M is co n stan t when re stricte d to U Q Q, then the cohomology of the bi-com plex M (U ..) is naturally isom orphic to H *( 7t(A U ..),M ). T h u s, the proposition follow s from Theorem 3 .8 by taking colim its indexed by the left cofinal su bcategory of H R R (X .) co n sistin g of U .. -> X .

su ch that M re stricte d to U Q Q is co n sta n t. ■

We ob serve that the hypothesis th at X .

be con n ected in P roposition

5 .9 is not n e ce ssa ry b e ca u se we can u se P rop osition 5 .2 to equate H *(X . ,M) with

n ae7T0 ( X . )

n a e n Q( X . )

H * ((X ? )e t ,M ).

H *(X a . ,M) and H * ((X .)e t ,M) with

6.

WEAK E Q U IV A LEN C ES, COM PLETIONS, AND HOMOTOPY LIMITS

In th is ch apter, we review the homotopy theory which has been employed in various ap p licatio n s of the e ta le to p o lo g ical typ e.

The role

of the co n stru ctio n s we con sid er is to enable us to obtain homotopy th e o retic information from the invariants of ( X .) e ^ con sid ered in C hapter 5. A fter review ing the definitions of various homotopy c a te g o rie s , we p resent the theorem of M. Artin and B . Mazur which provides n e ce ssa ry and su fficien t con dition s for a map to be a weak eq u ivalen ce in the prohomotopy categ ory [8].

We then proceed to re ca ll the Artin-Mazur pro-L

com pletion functor which en ables us to exclu d e from con sid eration homotopy information at sp ecified prim es’. Follow ing D. Sullivan, we next con sid er the Sullivan homotopy limit holim Su( ) which provides a c a t e ­ g o rical in verse limit for ce rta in p ro-ob jects in the homotopy categ o ry [69]. F in a lly , we con sid er the co n stru ctio n s A. K. B ousfield and D.M . Kan [13].

(Z/Q ^C

) and hoHm( ) of

T h e se ^-completion and homotopy

limit functors have the sig n ifica n t advantage of being “ rig id ” in the se n se that they take valu es in (s . s e t s * )

rather than in the homotopy ca te g o ry .

As we s e e , th e se co n stru ctio n s often provide a rigid version of the homotopy th eo retic co n stru ctio n s of Artin, Mazur, and Sullivan. We re ca ll that a map f : S. -» T .

in (s . s e ts ) is said to be a w eak

e q u iv a le n c e if the geom etric realizatio n of f is a homotopy eq u ivalen ce. We let

H,

the homotopy ca tego ry , denote the categ o ry obtained from

(s . s e ts ) by formally inverting the weak e q u iv a le n ce s. Sim ilarly, obtained from (s . s e t s * ) eq u iv alen ces.

We let

H*

by inverting pointed maps which are weak

(s. s e t s * c ) denote the ca te g o ry of pointed,

51

is

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

52

con n ected sim p licial s e ts and su b category of K * ).

K * c denote its homotopy categ o ry (a full

T hu s, a map in (s . s e t s * c ) is invertible in K *c

if and only if it induces an isomorphism on homotopy groups.

Although we

sh all not require their co n stru ctio n , we a ls o mention the co n stru ctio n of H o(pro-(s. s e t s * ) ) ,

a homotopy categ o ry of p ro-sim p licial s e t s , by

D .A . Edw ards and H.M . H astin gs [28]. We re ca ll that the n-th co sk e le to n functor, c o s k n : ( s . s e t s * ) -> (s . s e t s * ) ,

is the right adjoint of sk n( ) , 1 .3 ).

Unlike

sk n( ) ,

the n-sk eleton functor (a s in Definition

c o s k n( ) induces a functor c o s k n : K * ^ K* ,

b e ca u se for any pointed, con n ected sim p licial s e t T . T.

co sk n(T .) induces isom orphisms

and the homotopy groups

the natural map

^ ( T . ) -» 77'^ c o s ^n

^or k < n

77^(coskn T .) are zero for k > n . T his o b serv a­

tion permits us to give the following definition. D E F I N I T I O N 6 .1 .

to

The functor

# : ( s . s e t s * ) ^ (p ro -s . s e t s * )

sending T .

# (T .) = {c o s k n T . ; n > 0 ! exten d s to a functor # : pro-K*

by sending i s i j i e l !

pro-K*

to ic o s k n S1. ; (n ,i) e N x I } .

A map f :{ S * !- > { T ? ! in

pro-H* is said to be a w eak e q u iv a le n c e in pro-K*

if # (f ): #isM -> #\t I !

is an isomorphism in p ro -K *. ■ The introduction of weak eq u iv alen ces in pro-K* following theorem . (s . s e t s * c ) ,

is ju stified by the

As in Chapter 5 when we con sid ered o b je cts in pro-

for any iS1! e pro-K *c

we define

^ ( { s i i ) = { ^ ( S i ) ! and

H*(isi!,M) = colim H *(S! ,M) for any ab elian lo ca l co e fficie n t sy stem M on iS.1! (given by an eq u iv alen ce c l a s s of homomorphisms Aut(MQ) ,

77^(1 SM) -»

where MQ is a sta lk of M; the colim it is indexed by the left

final full su b category of I co n sistin g of th o se Aut(MQ) facto rs through

^ (i s M ) -> tt^S^)) .

j for which

^ ( i s l ! ) -»

6. WEAK EQ U IV A LE N C E S, COM PLETION S, AND HOMOTOPY LIM ITS

For any map

T H E O R E M 6 .2 (M. Artin and B . Mazur [8], 4 .3 and 4 .4 ).

f : {S.M -» |T^! in pro-H^ a .)

f

,

53

the follow ing a re e q u iv a len t.

rs a w eak e q u iv a le n c e in pro-K^ .

b .)

: 7t^({sM) -* 77^({T^!)

c .)

: 77i d s ! I) -> 77^

lo ca l c o e ffic ie n t sy stem

is an isom orphism for a ll k > 0 . is an isom orphism ; and for every a b elia n

M on{T ^ },

f* : H*({T^ S,M) -> H*(lSM,M)

is an

isom orphism . ■ We observed in C hapter 5 that the fundamental pro-group and the cohomology groups with ab elian lo ca l co e fficie n ts of (X . ,x ) e £ can be identified “ a lg e b ra ic a lly .”

Theorem 6 .2 im plies that th e se alg eb raic

invariants determ ine (X . ,x ) e ^ up to weak eq u ivalen ce in pro-K^ . As a first ap plication of Theorem 6 .2 in conjunction with Chapter 5 , we provide the following co ro llary . L et X. ,x

C O R O L L A R Y 6 .3 .

b e a p o in ted , c o n n e c te d , lo ca lly n o etherian

sim p licia l s c h e m e and let H R (X. ,x ) (X . ,x )ht H R (X. ,x )

,x)

-»

.

d eterm in es a w eak e q u iv a len c e

in p ro-X * , (X . ,x ) ht -> (X . ,x )e t . P roof. final.

We verify that the forgetful functor Let

U .. ,u

H R (X. ,x ) -> H R (X .) is left

be a pointed hypercovering of X . , x and let f ,g :U ..- > V ..

be two maps of hypercoverings of X . . Then extend to k : Spec 0 ® A [1 ] -» X Q b ecau se

f*(u ), g * ( u ) : Spec

-> VQ Q

VQ over the co n stan t homotopy Spec f}® A [l]

VQ x Spec 1) is co n tra ctib le (s e e the proof of Prop osition ' xo

3 .6 ).

Thus

h :H .. -» U ..

(k,u) : Spec H -» H .. of f,g

By C orollary 5 .7 ,

is a pointing of the left equalizer

with h^(k,u) = u . j : (X .

-»(X . ,x ) et

isomorphism on fundamental pro-groups.

in pro-H^c

induces an

By Theorem 6 .2 , it su ffice s to

E T A L E HOMOTOPY O F SIM PLIC IA L SCHEMES

54

verify that j induces an isomorphism j * :H * ( ( X . ,x ) e t ,M) ->H*((X. ,x )^ ,M ) for any ab elian lo c a l co e fficie n t sy stem

M on (X . ,x )e j. . Using the

cofin ality of HR(X. ,x ) -> H R (X .) , we identify colim

j*

H *( tt(A U ..),M ) -> colim

H RR(X.)

with the map

H *( tt(AV. .),M )

HR(X.)

which w as proved to be an isomorphism in C orollary 4 .6 . ■ If G is a group and

L

a s e t of prim es, the pro-L com pletion of G

is the system of fin ite, L -to rsio n quotient groups ( i .e ., only primes in L divide their orders) of G . By ch oosin g a sm all left final su bcategory

I

of the categ o ry of a ll su ch quotient homomorphisms, we obtain a pro-group (G )L : I -» (fin ite L -g ro u p s).

A s is ea sy to ch e ck , th is con stru ction

exten d s to a functor ( ) L : pro-(groups) -» pro-(finite L -groups) . The following theorem of M. Artin and B . Mazur provides the analagous con stru ction for p ro -K ^ . T H E O R E M 6 .4 (M. Artin and B . Mazur [8], 3 .4 and 4 .3 ).

of prim es.

T h en the in clu sio n p ro -L K s|eC -* pro-H

L et L

be a set

has a left adjoint

( ) L : p ro -K ^ -» p ro-L H *c

w here L H^c

is the fu ll su b ca teg o ry of

co n s is tin g of sim p licia l

s e t s w hose homotopy gro u p s a re fin ite L -groups.

T h e map on fundam ental

pro-groups in d u ced by the ca n o n ica l map JsM -> (isM )L com pletion map for 77^ ({SM)

is the pro-L

for any {SM e p ro -H ^ . M oreover, if M is

any a b elian lo ca l c o e ffic ie n t sy stem on ( { s ! i ) L

w hose fib re s a re fin ite

L -groups, then the ca n o n ica l map in d u ces an isom orphism H *(({sM )L ,M) h

* ({ s M,m) . ■

As a corollary of Theorem s 6 .2 and 6 .4 , we conclude the following.

6. WEAK E Q U IV A LE N C E S, CO M PLETIO N S, AND HOMOTOPY LIMITS

C o r o l l a r y 6 .5 .

L e t f : i s ! ! -» !T^ \ b e a map in pro-K.

be a s e t of p rim es.

T h en (f ) L

55

an d let L

is a w eak e q u iv a le n c e in pro-K*

if and

only if a .) j(f*)L : (771({SM ))L -* (77'1(iT^ !))L

is an isom orphism ; and

b .) for ev ery a b elia n lo ca l c o e ffic ie n t sy stem M on {T ? !

w hose

fib re s a re fin ite L -groups an d w hich is r e p r e s e n te d by a map 77’1({T ^ }) -> Aut(M0) facto ring through H*(iSM,f*M) Proof.

T^l)

}))L , f* :

The validity of a .) and b .) whenever (f ) L

in pro-K * is immediate from Theorem 6 .4 . co efficien t system

is a weak eq u ivalen ce

C o n v ersely , any abelian lo ca l

M on {T^i has a su b-system

ML whose fibres are

the maximal L -torsio n subgroups of the fibres of M. k>0,

Hk(({T ? l)L ,ML )

B ecau se

},M) ->

is an isom orphism .

Hk( ( H i ) L ,M) b e ca u s e

ML on ( { t ] S ) L

Furtherm ore, for any

({T ? I)l

is in pro-L K *c .

is a colim it of lo ca l co e fficie n t sy stem s with

finite L-groups as fibres and b e ca u se

H *((!T ^ i)L , ) commutes with

directed co lim its, we conclude using Theorem s 6 .2 and 6 .4 that a .) and b .) imply that ( f ) L

is a weak eq u ivalen ce in p ro-K ^ . ■

We now con sid er the Sullivan homotopy limit functor,

hoHmSu( ) ,

which en ables us to a s s o c ia te a sin g le homotopy type to ( ( X .) e t) L in Pr° - L K* c ‘ T H E O R E M 6 .6 (D. Sullivan [6 9 ], 3 .1 ).

prim es.

L et

P

d en o te the s e t of a ll

T h ere e x is t s a functor holimSu( ) : pro-P K *c -> K *c

together with a natural transformation holimSu( ) -> id : pro-P K *c n>pro-K*c ch a ra cteriz ed by the property that this natural transformation in d u ces a b ijectio n Homj/ (S. , holimSu( ! ' r i !)) - Horn *

j( (S. ,{T ? I ) = lira Homr, (S. , T?) * 5k

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

56

for ev ery S. holim Su(ST^ i)

in H^c

and { t ! ; j e ] \ e p r o - P H ^ .

We c h o o s e

to be a Kan co m p lex for ea ch iT^ \ e p r o - P K ^ .

for any s e t of prim es L

M oreover,

the Sullivan pro-L com pletion functor

holimSu( ) ° ( h as the property that if T .

) ^ : p ro -H ^ ->

is a pointed, c o n n e c te d sirn p licia l s e t with

fundam ental group so lv a b le of fin ite type and fin ite ly g e n e ra te d h ig h er homotopy gro u p s, then the ca n o n ica l map T . -> holim Su( ( T . ) L )

in d u c e s an

isom orphism in cohom ology H*(hoHmSu( ( T . ) L ),A ) for any fin ite a b elia n L -group A

and isom orphism s

77’*(holimSu( ( T .) L )) ^ 1

H *(T . ,A )

lim ((77.(T.)L )

0 . ■

1

x ) e j-L )*

^ (h o li m ^ C X . ,x ) e t L )) =

Unfortunately, Theorem 6 .6 does not in general enable

us to determ ine either the higher homotopy groups or cohomology groups of holim Su((X . ,x ) e t L ) .

N e v e rth e le ss, the following immediate co ro llary of

Corollary 6 .5 and the isomorphism

tt. (holim Su( *

T . H *(S. , Z /£)

id -» (Z /£ )

induces an isomorphism f* : H *(T . ,Z /£ ) —*

if and only if (Z i/t)oQ( i )

is a homotopy eq u ivalen ce.

over, ( Z / Q ^ f ) is a Kan fibration whenever f : S. -» T . b e ca u se

(Z /E )^ * ) = * ,

( Z / Q ^ S .) determ ines

where

*

More­

is s u rje ctiv e ;

is the triv ial pointed sim p licial s e t,

is a Kan com plex for any sim p licial s e t (Z / t ) J i

( ) such

S.

and

(Z / f ) J i

)

) :'(s - s e t s *) ^ ( s - s e t s *) ([1 3 ], 1 .4 .2 ).

The B ousfield-K an homotopy (in v erse) limit functor holim( ) : ( s . s e t s 1) is a functor on the ca te g o ry , s e ts for any sm all categ o ry naturality with re sp e ct to

(s . s e t s 1) ,

of “ I-diagram s” of sim p licial

I ( i .e ., functors from I to (s . s e t s ) ) . I en ab les one to extend

functorial with re sp e ct to s tr ic t maps: F : J -> (s . s e ts )

(s . s e ts )

The

holim( ) to be

if a : I -> J , G : I -> (s . s e ts ) , and

are fu n ctors, then a natural transform ation F °a -> G

determ ines a map holim (F) -> h olim (G ). A natural transform ation

F -> G

of I-diagrams with the property that F ( i ) -> G (i) is a homotopy eq uivalen ce for each

i e I determ ines a homotopy eq uivalen ce

provided that each a : I -> J

F (i)

holim (F) -> holim(G)

and G(i) are Kan co m p lexes.

M oreover, if

is a left final functor betw een sm all left filtering c a te g o rie s and

G : J -> (s . s e t s )

is such th at G(j) is a Kan com plex for each

holim(G) ->h olim (G °a)

j e J , then

is a homotopy eq u iv alen ce ([1 3 ], X I .9 .2 ).

homotopy limit of Kan com p lexes is again a Kan com plex.

The

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

58

In the following theorem , we summarize se v e ra l more useful properties ) and holim( ) . We re ca ll that a nilpotent sim p licia l s e t is

of ( Z /T ) ^

a pointed, con nected sim p licial s e t with nilpotent fundamental group which a c ts nilpotently on the higher homotopy groups. T h e o r e m 6 . 8 (A .K . B ousfield and D.M. Kan [13], 1 .7 .2 , V I .6 .2 , V II.5 . 2 , X I .7 .1 ).

For any pair of s im p licia l s e t s

S.

and T.

and any prime i ,

the ca n o n ica l map ( Z / Q ^ S . x T .) -> ( Z / Q ^ S .) x ( Z / Q ^ T . )

is a homotopy

eq u iv a le n c e with a natural right in v e rs e . If S. e (s . s e ts ^ c ) then S. -» ( Z / Q ^ S . )

e ith er is nilpotent or has fin ite fundam ental group, in d u c e s an isom orphism H ^ Z / ^ ^ S . ) , Z /£) ->

H*(S. , Z / £ ) . For any I-diagram F : I -> (s . s e t s %) F (i)

is a Kan co m p lex for ea ch

i e I,

with the property that

there e x is t s a s p e c tra l s e q u e n c e

of coho m ological type = lims |7rt (F i)| = > 77t_ s (hoH m (F)) ( w here lim °( )

is the in v e rs e limit functor,

0 < s < t

lim ^ ) :(g rp s*) -> grps)

first d eriv ed functor, a n d lims ( ) : (ab. g rp s1) -> (ab . grps) •

•

•

•

its s-th d eriv ed

«

functor) w hich c o n v e rg e s in p o sitiv e d e g r e e s p rovided that lim for a ll s ,t

with 0 < s < t . B

its

1

n

4-

E ' =0

r

The following proposition will usually be applied in the s p e c ia l c a s e that each component of each which c a s e

S? and T^ has finite homotopy groups (in

lim^ 77-m(Sa SM) = 0 for all k , m > 1 ).

erality of d iscon n ected

s ! and

We require the gen­

T^ for our d iscu ssio n of function com ­

p lexes in Chapter 11. P R O P O S I T I O N 6 .9 .

p ro-(s. se ts ^ ) S] , T^

L e t f : \Sl ; i ell -> {T I ; j 0 ,

^ holim { a T;! i .

homotopy e q u iv a le n c e ; if, in addition, c ien tly large k , P roof.

then holim(a f ) is a 1) = ^ ^or a ^

lim^

then holim |a SM and holim { a T^l a re c o n n e cte d .

We employ the co n stru ctio n s of [13].

Yn = T o tn( 11*1 T^ i) , so that

Let

X n = T o tn( I I * {s i l ) and

holim Isi i = lim SXn ! and holim IT^ I = lim {Yn S.

The s tric t map f induces a map from the “ first derived homotopy se q u e n c e s ” of [13], I X .4 .1 for iX n S

-

- "

2X ^

l

-

1Lmm W

* X n*> - " l X m } - - l X ^ l

-

m>° to th o se for {Yn i , where

n[ X^1) = image O i(X m+1)

[13], X I .7.1 and the isomorphism

lims n J l s i 1)

conclude that f induces an isomorphism and each

i > 0.

77j(Xn)

^ ( X ^ ) ) . Using

lims ^ ( { t ! !) , we ^iO ^) for e a ch

n

C onsequently, the Milnor e x a c t seq u en ce implies that

f induces isomorphisms

77-(lim X n)

77*(lim Yn) for i > 0 ,

s o that

holim(f ) re s tric ts to an eq u iv alen ce on con nected com ponents. The fa ct that

holim (f) = II holim(a f ) a or

is immediate from the o b serva-

tion that any map A ->II*lsiS fa cto rs through II* {a siS -^ II*{sil for some a.

F ix some clett, and define

'S ? = h olim \a sY\ and i/I

Then the natural maps

' T l = holim Sa T ? i . j/J

a S? -> ' s\ and aT\ -> ' t I are homotopy eq uiva­

le n ce s ([1 3 ], X I .4 .1 ), so that

holim(a f )

is a homotopy eq uivalen ce if and

only if holim( ' f ) : holim \ 'SM ^ holim i 'T ? l is a homotopy eq u ivalen ce.

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

60

Observe that a ch o ice of b a se point of holim !a SH provides {

' f : { ' S 1.}

with the stru ctu re of a s tr ic t map in p ro-(s. s e ts ^ c ) which is an

isomorphism in pro-K^ . C onsequently,

holim( ' f ) : holim !

' s\\

-> holim ! 'T ? ! re s tric ts to a

homotopy eq u ivalen ce on con nected com ponents a s s o c ia te d to any b ase point of h o lim !a S* I.

If limm 7rm( ! a S}i) = 0 for a ll m > 0 , then we co n ­

clude using the above first derived homotopy seq u en ces that { * } = tTq 'Y ^ '* for a ll

m > 0.

; hence

=

T his im plies that h olim (f) induces a

b ijectio n on con n ected com ponents, with 7r0(holim ! ' ! )

tt^

h o lim ('f )

770(holim { 'SM) =* lim 1 7Ti ' X m e*

is a homotopy eq u ivalen ce.

If

lim^- 1 77-^({a SM) = 0 for k su fficien tly larg e, then th ese first derived homotopy seq u en ces imply th at \tt^( X n)S is M ittag-L effler so that holim !'S.M 0 , i h o lim (Z /£)n(T .) eq uivalen ce for any

T. e

. ■

is a homotopy

7.

FIN IT EN ESS AND HOMOLOGY

In this ch apter we co n sid er noeth eria n sim p licia l s c h e m e s ( i .e ., sim p licial sch em es which are noetherian in e a ch dim ension).

As we

verify in C orollary 7 . 2 , the e ta le to p o lo g ical type of su ch a noetherian sim p licial schem e is weakly equivalent to a p ro-object in the homotopy categ ory of sim p licial s e ts which are finite in e a c h dimension.

In

Theorem 7 . 3 , we g en eralize to sim p licial sch em es the criterion of M. Artin and B . Mazur th at the homotopy pro-groups of the e ta le top ologi­ c a l type be pro-finite.

In Prop osition 7 . 5 , we show that pro-ob jects of

finite ab elian groups are anti-equ ivalent to torsion ab elian groups.

Under

th is an ti-eq u iv alen ce, the homology pro-groups with c o e fficie n ts in the dual of a sh eaf of co n stru ctib le ab elian groups (a s defined in Definition 7.4)

correspond to cohom ology.

A theorem of P . D eligne provides

exam p les of sim p licial sch em es with finite homology groups with various c o e fficie n ts. P ROPOSITION 7.1.

sch em e.

L et X. , x

b e a p o in ted noeth eria n sim p licia l

T h en the fu ll s u b c a te g o rie s n H R (X .) C H R (X .) , nHR(X. ,x ) C H R(X. ,x )

c o n sistin g of hyper co v e rin g s U .. ^ X . s ,t > 0

with Ug ^ noeth eria n for a ll

(“ noetheria n hy p er c o v e rin g s ” ) a re left filterin g and the in clu sio n

functors a re left final. P roof.

If U -> X n and V -> X n are e ta le with U and V noetherian,

then th ese maps are of finite typ e; thus, the fibre product of U -» X n X n in E t ( X .) ,

U is noetherian if and only if U is a

union of finitely many con n ected com ponents. prove for any pointed hypercovering V .. ,v -» U .. ,u su ch th at Vg t

T h erefore, it su ffice s to

U .. ,u -> X .. ,x

Us t

the e x is te n c e of

is the inclusion of finitely many

com ponents of Ug ^ for all s ,t > 0 and V .. ,v -> X .. ,x hypercovering.

We define

VQ Q

is a pointed

U Q Q to be the pointed inclusion of

finitely many com ponents of U Q Q which co v er X Q. P roceed in g inductively, we define

Vg t -> Ug t to be the in clu sion of those co n ­

n ected components of Ug t which are in the image of Vg l t or Vg t l under some d egeneracy map of U ..

together with finitely many co n ­

n ected com ponents of the fibre product of (cosk^ ^Vg 0

= X g = c o s k jV g ) . ■

As an immediate corollary of Prop osition 7.1 and C orollary 6 .3 , we obtain the following fin iten ess property of the weak homotopy type of (X . ,x )e t . C O R O L L A R Y 7 .2 .

L et X. ,x

sim p licia l s ch em e .

D e fin e (X .

homotopy type of X . , x ,

b e a pointed, c o n n e c te d , n o etherian in pro-H^ ,

the noeth eria n (e ta le )

to be

(X . ,x ) nfa = 77°A : nHR(X. ,x ) -> (s . s e ts ^ ) . T h en (X . ,x ) n^. is a p ro -ob ject in the homotopy ca tego ry of po in ted sim p licia l s e t s w hich a re fin ite in ea ch d im en sio n .

Furtherm ore, the

natural maps (X- >x )nht

(X - »x )ht

^X - ’x ^et

a re w eak e q u iv a le n c e s in pro-K^ . ■ We remind the reader that a schem e

X

is said to be geom etrically

unibranched if the in tegral clo su re of e ach of the lo ca l rings of X

(sta lk s

7. FIN IT E N E SS AND HOMOLOGY

65

of the stru ctu re sh e a f in the Z a risk i topology) is a ls o a lo c a l ring.

In

p articu lar, if e ach of th e se lo c a l rings is already integrally clo se d (as is the c a s e if they are regular lo ca l rin g s), then X

is g eom etrically

unibranched. The following theorem is an e a sy gen eralizatio n of a theorem of M. Artin and B . Mazur ([8 ], 1 1 .2 ).

We sh all find th is theorem p articularly

useful when we co n sid er function co m p lexes. T H E O R E M 7 .3 .

L et X. ,x

be a pointed, n o eth eria n sim p licia l s c h e m e

su ch that X n is c o n n e c te d and g eo m etrica lly u n ib ra n ch ed for ea ch T h en any pointed, noeth eria n h y p erco v erin g property that

A U . . ) , u)

U .. , u

is fin ite for e a ch

k > 0.

X. ,x

n > 0.

ha s the

C o n seq u en tly ,

77k ((X . ,x ) e ^.) is a pro-finite group ( i .e ., isom orphic in pro -(grp s) to a pro­

o b ject in the ca teg o ry of fin ite g ro u p s). Proof. any

F o r any pointed, noetherian hypercovering U .. , u -» X . , x and

n > 0 , Un -» X n is a pointed, noetherian hypercovering so that the

Artin-Mazur theorem im plies that (B e c a u s e U -> X

^ ( ^ ( b ^ )) is finite for e a ch

k > 0.

X n is geom etrically unibranched, any con n ected e ta le open

is irreducible.

th at th is implies that

The startin g point of their proof is the observation 7r(Un ) =

x Spec K ) , where

K is the field

' xn of fractio n s at the generic point of X n .) B e c a u s e e a ch

?KUn ) is co n ­

nected by P rop osition 5 .2 , we may apply the homotopy s p e ctra l seq uen ce of [12], B .5 , to con clu d e th at the homotopy groups of a ls o finite.

77-^(t7( A U . . ) , u )

are

C orollary 7 .2 now im mediately im plies th at ^ ( ( X . ,x ) j.) is

pro-finite for each

k > 0. ■

The n e c e s s ity of the hypothesis that ea ch

X n be con nected in

Theorem 7 .3 can be readily understood by examining the c a s e in which X. = S p ecK ® S 1 . Our approach to the homology of an ab elian co p resh eaf is motivated by C orollary 3 .1 0 .

We sh a ll find it su fficien t to co n sid er co p re sh e a v e s

66

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

which commute with finite d isjoint unions b e ca u s e we sh a ll re s tric t our attention to noetherian sim p licial sch e m e s; the reader should observe that the dual of a sh eaf does not commute with arbitrary d isjoint unions. D E F I N I T I O N 7 .4 .

Let X.

c o p re s h e a f on E t ( X .) property that F o r any

be a sim p licial sch em e.

is a functor

P : E t ( X .)

An a b elia n

(ab. grps) with the

P (U H V ) - P ( U ) e P ( V ) for any U ^ X n , V -+Xn in E t ( X .) .

i > 0,

the i-th hom ology pro-group of X .

abelian cop resh eaf

P

with valu es in the

is the pro-group

H ^ X . #P ) = Hi o P ( ) : H R (X .) -> (ab. grps) sending

U .. -» X .

to the i-th homology group of the bicom plex

P ( U ..) . ■

We sh all be e sp e c ia lly in terested in the following two exam ples of abelian co p re sh e a v e s. E t ( X .) ,

If M is a lo ca lly co n stan t ab elian sh eaf on

we define M ° :E t ( X .) -» (ab. grps)

by settin g M °(U) = ®M°(Ua ) , con n ected com ponents M restricted to

Ua

Ua

where the d irect sum is indexed by the

of U and where M °(Ua ) = M(Ua ) whenever

is co n sta n t and M °(Ua) = 0 otherw ise; we define

M °(U) ->M°(V) a s s o c ia te d to a map U -> V in E t ( X .) to be the map whose restrictio n to M °(Ua ) is given by the in verse of M(V^g) -> M(Ua ) , where Vjg is the component of V containing the image of Ua restricted to Ua If F F (V ),

if M

is co n stan t.

is an abelian presheaf on E t(X .)

such that F (U II V) = F (U ) x

then we define the dual c o p re s h e a f F v : E t ( X .) -» (ab. grps)

by F V(U) = H om (F(U ), Q /Z ) . If P : E t ( X .) -> (ab. grp s) tak es v alu es in the categ o ry of finite abelian groups (denoted -> H R (X .) im plies th at

(f. ab. grps) ), then the left finality of n H R (X .) H^(X. ,P )

is pro-finite for e ach

i > 0.

Such

67

7. FIN IT E N E SS AND HOMOLOGY

pro-finite abelian groups are not so unfam iliar, as we show in the next proposition. P

r o po sitio n

7 .5 .

L e t ( ) v = Hom(

,Q /Z ) .

T h en the functor

colim o ( ) v :(p ro -(f. ab. g rp s ))0 -» (tor. ab. grps) is an e q u iv a len ce of c a te g o rie s from the o p p o site ca tego ry of the ca tego ry of p ro -ob jects of fin ite a b elia n groups ( “ pro-finite a b elia n g ro u p s ” ) to the ca tego ry of torsion a b elia n gro u p s.

In pa rticu la r, {A- \ e pro-(f. ab. grp s)

is fin ite ( i .e ., isom orphic to a fin ite group) if a n d only if colim {A^i fin ite if and only if lim {A -l P roof.

is

is fin ite.

B e c a u s e any torsion abelian group A is the colim it of its finite

subgroups, b ecau se the categ o ry of finite subgroups of A is left d irected (where B

maps to C in this category if and only if B

and b ecau se

co n tain s

C ),

( ) v is a (co n trav arian t) involution of (f. ab. grp s), the

functor colim ° ( ) v is e s se n tia lly s u rje ctiv e . i elS -» !H j; j e ] I

T o prove colim ° ( ) v is faithful, co n sid er f,g : in pro-(f. ab. grps). for e a ch

j £J

Then f = g if and only if fj = gj

if and only if f J = g j

if and only if f^ = g j

in colim Hom(G-,Hj)

in colim Hom(H^,G^) for each

in Hom(Hj/, c o lim \G^\) for e a ch

j e]

j £J

(b e ca u se

Hj' is fin ite) if and only if colim fj' = colim g^ in Hom(colim {H ^l,colim {G^i). To prove c o l i m o ( ) v is fully faithful, co n sid er f : colim colim {G^i and w rite

1

f = lim f - with f •: colim lG-v ! . Since

fin ite,

fj facto rs through

G^

fin ite, th is image facto rs through some

is

the image

fa cto rs through some map f • ^

J>K

map determined by f ^ u , J fK

J

1

Gj£ -> colim iG^S. J

is

1

T hu s, J

be the

then the fa c t that colim ° ( ) v is faithful im-

plies that ifj^l: iGjJ -» {H jl is a w ell-defined map and that colim o ( ) v({fj/ !) = f .

J

of some G ^ -> colim {G ^ l;

G^ . If we le t f^ : lG -! -> HK

-»

fj

and sin ce

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

68

In p articu lar, the fa c t th at colim ° ( ) v is an eq u ivalen ce im plies th at colim {A^S = A is finite if and only if |A^| is isom orphic to a finite group A .

Since

(colim {A ^ i^ = lim { A j l , colim {A^S is finite if and only

if lim{A*S is finite. ■ We next u tilize the eq u ivalen ce of P rop osition 7 .5 to re la te homology to cohomology. schem e

We remind the reader that a sh eaf F

on a noetherian

(with the e ta le topology) is said to be c o n stru ctib le if there

X

e x is ts a finite c o lle c tio n of lo ca lly clo se d su bschem es disjoint union is

X

such that F

co n stan t with finite s ta lk s . c ia l schem e and F

re stricte d to e a ch

More gen erally, if X .

is a sh eaf on E t ( X .) ,

X-

then F

F (U ) is a finite s e t for any

whose

is lo cally

is a noetherian sim pli­ is said to be co n ­

stru ctib le if F n is co n stru ctib le on E t(X n) for e a ch co n stru ctib le, then

iX-S of X

n>0.

If F

is

U -* X n in E t ( X .) with

U noetherian. P R O P O S I T I O N 7 .6 .

F

L et X.

b e a noeth eria n sim p licia l s c h e m e , and let

b e a co n stru ctib le a b elia n s h e a f on E t ( X .) .

For ea ch i > 0 ,

th ere is

a natural duality isom orphism (colim o (

) v) (H -(X . , F v))

-

Hi(X . , F ) ,

so that H -(X. , F V) is fin ite if a n d only if H*(X. , F )

is fin ite.

M oreover,

if M is a locally co n sta n t c o n stru ctib le a b elia n s h e a f on E t ( X .) , for a ll i > 0

th ere a re natural isom orphism s (colim ° ( ) v)(H i(X . ,M0))

P roof. If U .. -> X . sh e a f, then

then

*

Hi (X . ,M) .

is a noetherian hypercovering and F

a co n stru ctib le

( F V( U ..) ) V is isom orphic to F ( U ..) s o that (H -(F V( U ..) ) ) V ^

H * (F (U ..)) for any i > 0 .

C onsequently,

colim (H i( F v(U ..) ) v) s* colim H ^ U . . ) ) where the co lim its are indexed by n H R (X .).

~

H ^X. ,F )

7. FIN IT E N E SS AND HOMOLOGY

69

If M is a lo ca lly co n sta n t, co n stru ctib le abelian sh eaf on E t ( X .) , then M °(U ..) is naturally isom orphic to MV(U ..) whenever U .. noetherian hypercovering su ch that M re stricte d to

is a

U Q 0 is co n sta n t.

C onsequently, colim (H i (M °(U --))V) “

colim (H i(Mv( U ..) ) v)

“

H ^ X . ,M) . ■

Using P rop osition 7 .6 , we immediately conclude the homology analogue of the sp e c tra l seq u en ce of P rop osition 2 .4

E s ,t = Ht (X s , F v s) = > Hs + t( X , F v) w henever X .

is noetherian and F

is co n stru ctib le .

We ca n a ls o co n ­

clu d e, for exam ple, the homology analogue of P rop osition 3 .7 a sse rtin g that H#(X . , F V) whenever U .. -> X .

~ H *(A U .. , F V)

is a noetherian hypercovering and F

is co n stru ctib le .

We conclude th is ch apter with the following theorem whose proof is an immediate con seq u en ce of P . D elig n e’s criterion for the fin iten ess of cohomology ([2 4 ], “ F in itu d e ” 1 .1 ) (extended to sim p licial sch em es using P rop osition 2 .4 ) and P rop osition 7 .6 . T H E O R E M 7 .7 .

w here R

L et X.

be a sim p licia l s c h e m e of fin ite type over Spec R ,

is a co m p lete d is c r e t e valuation ring with a lg eb ra ic a lly c lo s e d

re s id u e fie ld .

T h en for a ll i > 0 , H ^ X . , F )

for any co n stru ctib le a b elia n s h e a f F

an d H ^ X . , F V) are fin ite

on E t( X .) with sta lk s of order

in v ertib le in R . ■ The condition on the orders of the sta lk s of F n e ce ssa ry .

in Theorem 7 .7 is

F o r exam ple, H ^ S p e c k[x], Z /p ) is infinite whenever k is an

infinite field of c h a ra c te r is tic

p.

8.

COMPARISON O F HOMOTOPY T Y P E S

In Chapter 5 , we identified various homotopy invariants of the e ta le top o lo gical type

(X . ,x ) e j. in terms of alg eb raic in variants.

In Chapter 6,

we d escrib ed the hom otopy-theoretic co n te x t in which th e se invariants play a cen tral role.

We now proceed to employ th is m aterial to tra n sla te

various theorem s concerning e ta le cohomology groups and fundamental groups into theorem s concerning the homotopy type of (X . ,x )e j.. We begin by comparing in P rop osition 8.1 the homotopy type of (X . ,x )e £ to that of (A U .. ,u )e |., where U .. , u -» X . , x

is a pointed hypercovering.

The proof of P rop osition 8.1 is rep resen tativ e of the method of proof of e ach of the re su lts in this ch ap ter.

F o r a sim p licia l schem e X .

over the

com plex numbers, Theorem 8 .4 p resen ts the very useful com parison of the homotopy type of (X . ,x )e t with the homotopy type of its underlying sim p licial sp a ce with the “ c l a s s i c a l to p o lo g y .”

Prop ositions 8 .6 and 8 .7

obtain homotopy th eo retic co n clu sio n s from the proper b ase change theorem and smooth b ase ch ange theorem for e ta le cohom ology.

F in a lly , in

Prop osition 8 .8 , we in v e stig a te reductive group sch em es and their c l a s s i ­ fying s p a c e s . In this first proposition, we conclude that from a homotopy-theoretic point of view a sim p licial schem e may be replaced by one of its hyper­ cov erin g s.

The u sefu ln ess of this co n clu sio n lie s in the fa ct that the

hypercovering may c o n s is t of sch em es which are sim pler (being more lo ca l) than the original sim p licial sch em e.

F o r exam ple, this is the b a sic

observation underlying the d iscu ssio n of tubular neighborhoods of Chapter 15.

A somewhat weakened form of the following proposition was proved by

D. C ox in [18], IV .2.

70

8. COMPARISON OF HOMOTOPY T Y P E S

P R O P O S I T I O N 8 .1 .

L et X. ,x

71

b e a pointed, c o n n e c te d , lo ca lly n o etherian

sim p licia l s c h e m e , and let g : U .. , u -» X . , x

be a p ointed h y p er co v erin g .

T h en ge t : (A U .. ,u)e t - ( X . ,x )et is a strict map of p ro-(s. s e ts ^ c ) w hich is a w eak e q u iv a len ce in pro-H^ . In particular, for a ll i > 0 , g^ : 7t-((AU.. ,u ) t )

^ ( ( X . ,x ) t)

is an

isom orphism , and if th e s e pro-groups a re pro-finite, then (h o lim °S in g .( )o| |) ( g ) : holim (Sing.(|(AU.. ,u )e t |))->hoUm(Sing.(|(X. ,x )e t |)) is a homotopy e q u iv a le n c e (w h ere

| |: (s . s e ts ) -» (top. s p a c e s )

is the

geo m etric rea liza tio n functor and Sing.( ) : (top. s p a c e s ) -* (s . s e ts )

is the

sin gu la r functor a s d is c u s s e d in [42]). P roof.

By P rop ositio n s 3 .7 and 5 .9 ,

H * ((X .)e j.,M)

g : A U .. -> X .

H *((A U ..)e |_,M) for any abelian lo ca l co e fficie n t system

M on ( X .) e £ . By Theorem 6 .3 , to prove that in pro-K ^,

induces an isomorphism

ge j. is a weak eq uivalen ce

it su ffice s to prove th at g a lso induces an isomorphism

771((A U .. ,u)e t ) -» ^ ( ( X . ,x )e t ) .

By Prop osition 5 .6 , it su ffice s to prove

that g induces an eq u ivalen ce of c a te g o rie s any group G .

I1(X. ,G)

II(A U .. ,G ) for

As re ca lle d in the proof of P rop osition 5 .6 , th ere is an

eq u ivalen ce of ca te g o rie s

I1(XS,G)

n (U g ,G) for any

s > 0.

C o n se­

quently, the required eq u ivalen ce follows from Lemma 5 .4 and P roposition 5 .5 , which provide the interm ediate eq u iv alen ces < r i ( X 0,G); d. data > and < II(U a ,G); d. data >

II(X . ,G) I1(AU.. ,G ) .

Theorem 6 .2 now im plies th at ge ^ induces an isomorphism on homotopy pro-groups, w hereas P roposition 6 .9 im plies that (holim °S in g. ° | |)(g) is a homotopy eq uivalen ce (b e ca u se the natural map S. -» Sing. (|S. |) is a weak eq uivalen ce with Sing. ( |S. |) a Kan com ­ plex for any sim p licial s e t S. ). ■ Most ap p licatio n s of e ta le homotopy theory have u tilized the “ C ech to p o log ical t y p e / ’ A (X. ,x ) re|., a s defined preceding D efinition 4 .4 .

We

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

72

verify th at A (X. ,x )

^ has the sam e weak homotopy type as

for “ m o st” pointed sim p licia l sch em es L et X. ,

P R O P O S I T I O N 8 .2 .

X

X. , x .

b e a pointed, c o n n e c te d , lo ca lly n o etherian

sim p licia l s c h e m e , a n d let A ° ( X . >x )ret (a s d e fin e d a b ov e).

(X . ,x )e t

d en o te the C e c h top o logica l type

T h en there is a natural stric t map of p ro-(s. s e ts ^ c ) (X . ,x )et -> A o( X . ,x )ret

w hich in d u ces an isom orphism ^ ( ( X . >x )e t) ^

77’1(A o (X . ,x )re|.). Fu rther­

more, if X n is q u a si-p ro jectiv e over a n o eth eria n ring for e a ch n > 0 , then this map is a w eak e q u iv a le n c e in pro-K^ . P ro o f. The a sse rte d s tr ic t map is induced by the rigid analogue of cosk ^ *( ) : C (X .) -> H R (X .) . The fa c t that this map induces an isom or­ phism on fundamental pro-groups follow s d irectly from C orollary 5 .7 . Arguing a s in the proof of Proposition 5 .9 , we e a s ily conclude the natural v isomorphism H *(X . ,M) H*(A o (X . ,x ) re|.,M) for any lo ca lly co n sta n t, abelian sh eaf M on E t ( X .) . noetherian ring for e a ch

T hu s, if X n is q u asi-p ro jectiv e over a

n > 0,

then Theorem 3 .9 implies that (X . ,x > e t -

A o (X . ,x )re|. induces an isomorphism in cohomology with any ab elian lo ca l c o e fficie n ts .

C onsequently, for a pointed sim p licial sch em e, the

fa ct that (X . ,x ) e t ^ A ° (X . ,x ) re£ is a weak eq u ivalen ce in pro-H^ follow s from Theorem 6 .2 . ■ We next proceed to co n sid er sim p licial sch em es C ( i .e .,

X.

of finite type over

X n is of finite type over Spec C , where C denotes the com ­

plex numbers).

If X

com plex points of X finite type over

C,

is of finite type over C , then X to P is the s e t of with the usual (a n a ly tic) topology; if X . then x ! ° P

is of

is the sim p licia l s p a c e ( i .e ., the sim pli­

c ia l object of top ological s p a c e s ) with

(x t°P )n = X * °P .

We re ca ll that the g eo m etric rea liza tio n of a sim p licial s p a ce is the quotient of the top o lo g ical sp a c e

II n> 0

T . , |T. |,

Tn x A[n] by the eq uivalen ce

73

8. COMPARISON OF HOMOTOPY T Y P E S

relation

( t ,a ( x ) ) ~ (a (t), x) for any a :A [n ]-> A [m ]

in A (where Tn x A [n ]

is given the product topology and A[n] = lx = ( x 0,-*- ,x n) : Sx^ = 1, x- > 0 j c R n+1) We a lso re ca ll th at if S..

is a b i-sim p licial s e t and if In

a s s o c ia te d sirnp licial s p a c e , then

|A(S..)|

|S

|l is the

(the geom etric re a liz a tio n of

the diagonal sirnp licial s e t) is homeomorphic to

|in h> |Sn |!| (cf. [65], 1).

We now give a te ch n ica l lemma relatin g the singular cohomology of the geom etric realizatio n of a sirnplicial sp a c e

T.

to its sh eaf cohom ology.

In analogy with D efinition 1 .4 , we define the lo ca l hom eom orphism s it e L h (T .) as follow s.

As a ca te g o ry ,

homeomorphisms W -> Tn for some

L h (T .) has o b je cts which are lo ca l n > 0;

a map in L h (T .)

is a commuta­

tive square W ---------------

Z

Tn -------------►Tm with Tn

Tm a sp ecified stru ctu re map of T . ; a covering of W -> Tn

is defined to be a family of lo ca l homeomorphisms w hose im ages of W- in W co v er W.

{Wj->Wi over Tn

As in Definition 2 .3 , for any

i > 0 we define Hj;h(T . , ) : A bSh(T .) -» Ab to be the i-th right derived functor of the functor sending an ab elian sh eaf F

on L h (T .) to the kernel of the map d ^ - d * : F (T Q) -» F ^ ) .

L E M M A 8 .3 .

L et T.

b e a sim p licia l s p a c e with

paracom pact for

ea ch n > 0 .

For any lo ca lly co n sta n t a b elia n s h e a f M on L h ( T .) ,

a re natural isom orphism s H*h(T . ,M) w here Sing. ( T .) Sing t(Ts ) .

«

H *(A (S in g .(T .)),M )

«

H*(Sing. (|T.

|),M)

is the b isim p licia l s e t g iv e n in b id e g re e s ,t

by

th ere

74

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

P roof.

B e c a u s e every lo c a l homeomorphism W -> Tn is covered by a

family

fW- -» WS with e a ch

-> W -> Tn an open immersion, the ca te g o rie s

of ab elian sh e a v e s on L h (T .) and on the analogous s ite immersions are equivalent.

We define a sh e a f

O i(T .)

of open

on (3 i(T .) for any

p > 0 as the sh eaf a s s o c ia te d to the presh eaf sending an open immersion W ^ T n to

CP(Sing. (W),M) ([1 4 ], 1.7).

Then M -> S*M is resolution with

the property that (§P M )n is a c y c lic on G i(T ) for any I I I.l).

p,n > 0 ([1 4 ],

U sing the lo ca l homeomorphism analogue of Prop osition 2 .4 , we

con clu d e that H ^ ( T . ,M) is naturally isom orphic to H * (§ M (T .)). On the other hand, the natural map C *(S in g. (Tn),M) -» S M(Tn) induces an isomorphism in cohomology ([1 4 ], 1 .7 ) for each natural map of b icom p lexes phism in cohomology

n > 0 , so th at the

C *(Sing. (T .),M ) -» S M (T.) induces an isom or­ M ) • T hus, to com plete

H*(A(Sing. (T .)),M )

the proof, it su ffice s to verify th at the natural map A(Sing. (T .)) -> Sing. (|T. |) induces an isomorphism in cohom ology. paring the cohom ology of

T his is readily shown by com ­

|A(Sing. (T.))|

tion of the sim p licial s p a c e

!n

with th at of the geom etric re a liz a ­

|Sing. (T )|! with that of

|T.| . ■

The main ingredient of the proof of the following theorem is the “ c l a s s i c a l com parison theorem ” for e ta le cohomology proved by M. Artin and A. G rothendieck and the “ Riemann e x is te n c e theorem ” proved by H. Grauert and R . Remmert ([7 ], X I .4 .3 ).

V arious (w eaker) versio n s of

Theorem 8 .4 have been proved first by M. Artin and B . Mazur ([8 ], 1 2 .9 ), then by the author, R. Hoobler and D. R e cto r, and D. C ox. T H E O R E M 8 .4 ( C o m p a r i s o n T h e o r e m ) .

L et X. , x

sim p licia l sc h e m e of fin ite type over C , a s s o c ia t e d sim p licia l s p a c e (a s a bo v e).

be a po in ted , c o n n e c te d

and let X^°P, x

d en o te the

T h e re e x is t strict maps of

pro-(s. s e t s * c )

(X . ,x ) e t « L (X . ,x ) s et - L s in g .(| x J ° P ,x | ) su ch that r

is an isom orphism in pro-H^ , and (p )P

le n c e in pro-K^ , w here P

is a w eak eq u iv a ­

is the s e t of a ll prim es ( c f . C orollary 6 .5 ).

8. COMPARISON OF HOMOTOPY T Y P E S

P roof. We define

(X . ,x )g

A °Sing. ( ) o ( sending U .. -> X .

75

in p ro-(s. s e ts ^ ) to be the functor

)toP o A : H R R ( X . ) - ( s . s e t s * )

to the diagonal of the b isim p licial s e t

Sing. ( A u !? ^ ) .

The map r is the com position of the natural map 77: A(Sing. (A u !? ^ )) -> A(Sing. (x!" 0P)) (for any

U .. -» X . ) and the ca n o n ica l homotopy eq uiva­

len ce A(Sing. (X .)) -* Sing. (|xt°P|) (d iscu sse d in the proof of Lemma 8 .3 ). Using d escen t for principal homogeneous s p a c e s over X g° P ,

the argument

of [8 ], 10, ap p lies to show that II(X g0P,G) is equivalent to II(Ug0 P ,G ). As argued in the proof of P rop osition 8 .1 , th is im plies that II(A u t?P ,G ) is equivalent to I I ( x ! 0 P ,G ), which im plies that

^ (lA u l'P ^ u l) 2*

771(A(Sing. (u t? P ,u ))) is isomorphic to 7r 1(A(Sing. (X t° P ,x ))) ^ 77-1(| x t° P ,x | ). 77 induces an isomorphism of fundamental groups.

C onsequently,

As argued in C orollary 5 .8 , an ab elian lo c a l co e fficien t system

M on

A(Sing. (X t°P )) is in one-to-one correspon d ence with a lo ca lly co n stan t ab elian sh eaf on L h ( x t ° P ) . By Lemma 8 .3 and the lo ca l homeomorphism analogue of P rop osition 3 .7 ,

77 a ls o induces an isomorphism in cohomology

with ab elian lo ca l c o e ffic ie n ts . len ce for any

U ..

T herefore,

77 is a weak homotopy equiva­

in H R (X .) , so that r is an isomorphism in

X.

Pro-ft* • The map p is induced by the natural transform ation A oSing. ( ) ° ( defined by sending a : A[k] ->

)^0 P ° A ^ 77°A (A(Sing. (A U .t0 ^)))^ to the co n ­

nected component in (tz(AU ..))jc containing the image of a . T o prove that (p )P is a weak eq u iv alen ce in p ro-K ^, we facto r p a s the com posit ion S o y o ^ - l : (X . ,x )s>et - (X . ,x ) s i h - (X . ,x ) £h -> (X . ,x ) et defined as follow s. the s ite

E t(X .)

Let

H R R (X ?°P) be defined a s in P rop osition 4 .3 with

rep laced by L h (X t°P ) . Define (X . ,x )g g^ to be

E T A L E HOMOTOPY OF SIM PLIC IA L SCHEMES

76

A oSing. ( )°A :H R R (X ^ ° P ) -> (s. s e ts ^ ) and define

(X . ^ x)^

77o A : HRR(X^°P) -> ( s . s e t s . Define £ : (X . ,x )s>01

to be

(X . ,x ) s>et to be

the isomorphism in pro-K^ (a s argued ab ove, both are isom orphic to Sing. (|xJ°P,x|) ) determined by the forgetful functor

H R R (X .) ^ H R R (x t°P ).

Define y e x a c tly as we defined p , and define 8 to be a ls o induced by T hen, p ° /3 = < 5 ° y :(X . ,x )s

the forgetful functor.

As in P rop osition 5 .6 , we identify s o that y

tz^

-» (X . ,x )e t .

X . j X )^ ) with ^ ( I X ^ ^ x l ) ,

induces an isomorphism on fundamental pro-groups. As in

Prop osition 5 .9 , we identify

H ^(X^°P,M ) with H *((X . ,x)g^,M) for any

lo cally co n stan t ab elian sh eaf on L h (x !:0P ) , s o th at Lemma 8.3 implies that y induces an isomorphism in cohomology with ab elian lo ca l c o e fficie n ts.

T h u s,

y is an isomorphism in pro-M^ .

F in a lly , the Riemann e x is te n c e theorem and the usual d e sce n t argument relatin g

II(X^0 P,G) to < I ^ X ^ P .G ); d. data > and II(X. ,G) to < II(X 0,G);

d. data > imply that 8

induces an isomorphism on the

of the fundamental pro-groups.

pro-P com pletions

By the c l a s s i c a l com parison theorem and

P rop osition 2 .4 , we con clu d e that 8

induces an isomorphism in cohomology

with ab elian lo ca l co e fficie n ts w hose s ta lk s are finite.

T hu s, ( ( Z /£ ) J S i n g . (|x‘ °P ,x | )).

P roof. By P rop osition 6 .1 0 and C orollary 7 .2 , it su ffice s to prove that p and t induce homotopy eq u iv alen ces holimSU((X . , x & ) - holimSU((X . , x £ e t) - hoHmSu(Sing. |X?°P,x|£) .

T his follow s immediately from Theorem 8 .4 and C orollary 6 .7 . ■

77

8. COMPARISON OF HOMOTOPY T Y P E S

We next provide the homotopy th eo retic v ersion of the proper b a se change theorem in e ta le cohomology ([7 ], X II.5 .1 ).

We re ca ll th at a s trict

h e n s e l lo ca l ring is a lo ca l ring with a sep arab ly c lo se d residue field which s a tis f ie s H e n s e l’s lemma ([5 9 ], 1.4).

Such a lo ca l ring R has the

property that (Spec R )et is co n tra ctib le . P R O P O S I T I O N

L et

8 .6 .

R

b e a stric t h e n s e l lo ca l ring, and let

a c o n n e c te d sim p licia l s c h e m e over Spec R proper for ea ch

n > 0.

re s id u e fie ld of R ,

be

X .

su ch that X n -> S p e cR

is

L e t K be a sep a ra b ly c lo s e d fie ld containing the

an d let i : Y . -> X .

b e d e fin e d by Y = X n n

For any geo m etric point y

of Y Q,

x

n

Spec K.

S p ec R

the strict map of p ro-(s. s e ts ^ c )

i ' (Y- ,y )e t - ( X - >y)et is su ch that ( i ) P

is a w eak e q u iv a le n c e in pro-K^ , w here P

of a ll p rim es. C o n s e q u e n tly , for any prime £ ,

i

is the s e t

in d u ces a homotopy

e q u iv a le n c e i:h o li m o (Z /£ )oo((Y . ,y )e t) -> holim ° ( Z / £ ) oo((X . ,y )e t ) . Proof.

The proper b ase ch ange theorem im plies that i g induces an

eq u iv alen ce of c a te g o rie s

I1(YS ,G)

II(X S,G) for any finite group G .

By Lemma 5 .4 and P rop osition 5 .6 , th is im plies that

i induces a b ijection

Hom(77'1(X . ,x ),G ) -> Hom(771(Y . ,y ),G ) for any finite group G .

Moreover,

the proper b ase change theorem and P rop osition 2 .4 imply that i induces an isomorphism H *(X . ,M)

H *(Y . ,M) for any lo ca lly co n sta n t, co n ­

s t r u c t i v e ab elian sh eaf M on E t ( X .) . Prop osition 5 .2 . imply that

In p articu lar, Y .

is con nected by

By P rop osition 5 .9 , th e se cohomology isomorphisms

i induces an isomorphism

ab elian lo c a l co efficie n t sy stem

H * ((X .)e t ,M) -» H* ( ( Y . ) e t ,M) for any

M on ( X .) e |. with finite fib res.

By

Corollary 6 .5 , th is im plies that ( i ) P is a weak eq uivalen ce in p ro-K ^ . C onsequently, P rop osition 6 .1 0 and C orollary 7 .2 imply that holim ° (Z/Q oo((Y- ,y )e t) -> holim o (Z /£ )oo((X . ,y )e t ) is a homotopy eq u iv alen ce. ■

78

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

The proof of P rop osition 8 .6 ap plies a s well to prove the following theorem provided we employ the proper, smooth b a se change theorem ([7 ], X V I.2 .2 ) rather than the proper b a se change theorem . PROPOSITION 8 .7 .

L et R

b e a s trict h e n s e l lo ca l domain, and let X .

b e a c o n n e c te d sim p licia l s c h e m e over R proper and smooth for ea ch

n > 0 . L et

the stru ctu re map X Q ^ Spec R ,

s u c h that X n -> Spec R

e : Spec R

let x : Spec

is

X Q b e a se c tio n of

-> S p e cR -> X Q b e a

geo m etric point over the c lo s e d point of S p e cR , and let z : 12z -> S p e cR -> X Q b e a geo m etric point over the g e n e r ic point of S p e cR

(s o that any

e ta le neighborhoo d U -> X Q of x

is a ls o an eta le n eigh bo rh o o d of z ).

L e t j : Z. -» X .

= X

b e d e fin e d by Z

11

sep a ra b ly c lo s e d fie ld containing R .

x S p e c F , w here F 11 Spec R T h en j

is any

in d u c e s s trict maps in

p ro-(s. s e ts ) a n d pro-K ^ j : ( Z .) et - (x -)e t - j : (z - >z )ht - (X - ’x \ t su ch that ( j ) L

is a w eak e q u iv a le n c e in pro-M^ , w here L

is the s e t of

a ll prim es e x c e p t the re s id u e ch a ra cteristic of R . C o n seq u en tly , for any I eL , j

in d u ces a homotopy e q u iv a le n c e j : holim ° ( Z A ) J ( Z . ) e t) - holim o ( Z / £ ) J ( X . ) e t) . ■

We conclude this ch ap ter with the following s p e c ific com parison theorem which has proved useful in many ap p licatio n s (s e e , for exam ple, the d iscu ssio n of C hapter 9 ).

The reader can co n su lt [40] for an e x p licit

d iscu ssio n of G ^ and its cohom ological p roperties. PROPOSITION 8 .8 . G

L e t G(C) b e a co m p lex red u c tiv e L ie group, a nd let

be an a s s o c ia t e d C h ev a lley in tegra l group s c h e m e . L e t F

the a lg eb ra ic clo s u re of the prime fie ld F p , of F

let R

den o te

b e the Witt v ecto rs

(a co m p lete d is c r e t e valuation ring with re s id u e fie ld F ), and let

R -» C b e a ch o s e n em b ed d ing. containing F

T h en for any a lg eb ra ic a lly c lo s e d fie ld k

the b a s e ch a n g e maps BG^ -> BG p -> B G R et -> Sing.(B G (C )) w hose pro-L L = P -{p }.

co m p letio n s a re w eak e q u iv a le n c e s in pro-H^ , w here C o n seq u en tly , for any prime £ / p ,

th e s e maps d eterm in e a

chain of homotopy e q u iv a le n c e s betw een h o U m o (Z /l)oo((B G k )e t)

and

(Z /E^oSing.C B G C C )) .

P ro o f. The fact that the b ase change maps isom orphisms in Z /m

Gk -» Gp -> GR B G F -» B G R (B G R)et S in g.(B G (C )) have

pro-L com pletions which are

weak eq u ivalen ces is given by Theorem 8 .4 (and the homotopy eq uivalen ce Sing.(|BG^?P|)->Sing.(BG(C)) ). P roposition 6 .1 0 and C orollary 7 .2 now imply that the chain of maps ( 8 .8 .1 ) determ ines a chain of homotopy eq u iv alen ces relatin g holim ° ( Z / £ ) 00((B G k)e j.) and

(Z /£ ) oo°S in g .(B G (C )).

9.

APPLICA TIO N S TO TO POLOGY

In th is ch ap ter, we co n sid er two c la s s e s of top o lo g ical ap plications to the theory we have developed.

In Theorem 9 .1 , we present (a modified

version o f ) D. S ullivan ’s proof of the Adams C o njectu re.

T his is followed

by an infinite loop s p a c e version of the Adams C onjecture (Theorem 9 .2 ) whose proof requires the rigidity of the e ta le to p o lo g ical type and the B ousfield -K an co n stru ctio n s.

Theorem 9 .3 p resen ts a method of c o n stru c­

tion of maps of lo calized cla ss ify in g s p a c e s of L ie groups which are not induced by homomorphisms, w hereas Theorem 9 .5 d escrib es a re la tiv iz a tion of th is con stru ctio n to homogeneous s p a c e s .

T h ese co n stru ctio n s

u tilize p ositiv e c h a ra c te ris tic alg eb raic geometry. The reader will observe that a ll of the ap p licatio n s of th is chapter involve the study of sim p licial sch em es. In studying the sta b le homotopy groups of sp h eres, one employs the J ‘homomorphism J : O -» O ^ S 00 given by sending a : Rn -» Rn in On to the restrictio n of a ,

J ( a ) : S n_1 -> Sn_1

in Qn“ 1Sn -1 . In a se rie s of

papers, J . F . Adams determined the order of the image of J * : 77^(0) -> 00S°°) = tt^ (S °) up to 2-to rsio n [2].

While in v estig atin g this

J-homomorphism, Adams was led in [1] to a dram atic co n jectu re (the “ Adams C o n jectu re ” verified in Theorem 9 .1 ) concerning the generalized J-homomorphism con sid ered by M. F . Atiyah J : KO(X) -> J ( X ) (sending a real v ecto r bundle to its a s s o c ia te d sp h erical fibration) for a finite com ­ plex

X

[9], w hose solution for sp h eres com p letes the determ ination of

im (J* )C 4 ( s ° ) . In an influential paper [61], D. Quillen outlined a proof of the com plex K-theory analogue of the Adams C onjectu re.

T h is outline employed e ta le

homotopy theory, thus su ggestin g that the formalism introduced by M. Artin 80

81

9. APPLICATIONS TO TOPOLOGY

and B . Mazur to study a b s tra ct alg eb raic v a rie tie s might be “ turned around” so a s to u tilize alg eb raic geometry in the study of alg eb raic topology.

D. Q u illen ’s outline w as com pleted by the author in his th e sis

(cf. [29]); prior to the publication of th is com plete proof, Quillen in [63] provided an entirely different proof of the Adams C onjectu re based on the e arlier work of Adams and a technique of “ approxim ating” the orthogonal group by finite groups (s e e Chapter 1 2).

Independently, D. Sullivan pro­

vided a proof of the Adams C onjectu re which a ls o used e ta le homotopy theory ([6 9 ]); this proof differed in many re s p e c ts from Q u illen ’s outline, e sp e cia lly in that it does not involve alg eb raic v a rie tie s in c h a ra c te ris tic p > 0. We begin with an outline of S u llivan ’s proof of the Adams C onjectu re. The reader should re ca ll that the Adams operation

9*9 on (re a l) K-theory

KO(X) is determined by sending a bundle E -> X to the q-th Newton polynomial in the e x te rio r powers

A^E -> X

of E

X

(so that ^ ( E ->X)

is represented by a virtual bundle). T H E O R E M 9 .1 (Adams C onjectu re).

L e t J : BSO -» BSG re p r e s e n t the

(rea l) J -homomorphism, w here SO = USO(n)

is the in fin ite s p e c ia l

orthogonal group and w here SG = USGn with SGn the monoid of o rien ted s e lf-e q u iv a le n c e s of Sn . L e t ¥ 9 : BSO -> BSO re p re s e n t the q-th Adam s operation on (rea l) a lg e b ra ic K -theory, som e q > 0 . Jo ¥ 9 ,

T h en

J : BSO -> BSG

d eterm ine hom otopic maps of Z [ l / q ]

lo ca liza tio n s

Jo1*q ~ J : ( B S O ) 1 / q - ( B S G ) 1/q .

P roof (sk e tch ).

B ecau se

homotopy groups,

BSG is a simply co n nected s p a ce with finite

(B S G ). .

is homotopy equivalent to

\ where (

= holim1Su( ) ° (

II

(BSG)p,

ir* )£ . B e c a u s e

facto r through BSO -> (B SO )£,

J °*Pq , J : BSO -> BSG - (BSG)'^'

it su ffice s to prove that

82

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

J o ip q ,

are equal in

J : (BSO)g - (BSG)g

. We re c a ll that there e x is ts a g alois automorphism

o e G al(C ,Q ) su ch that

■An = V ° e t '

;

- ((BSOn ,c)et^ - « BSOn,C)et)'g - (BSO(n))'£

s ta b iliz e s (with re s p e c t to n ) to determine

^

: (BSO)£ -> (B S O )p

6n is determined by the two right-m ost arrows of P rop osition 8 .8 .

where C o n se ­

quently, it su ffice s to prove that J o*An > J :( B S O ( n ) ) - - .( B S G n_ 1) are equal in

for e a ch

n > 0.

One v erifies th at maps in

into (BSGn_ 1)^' are equivalent to fibre

homotopy c l a s s e s of fibrations with fibres determ ines the

(Sn - 1 ) p

where X -> (B S G ^ P g

(Sn _1)~ fibration given by the pull-back of the universal

fibration B ((S n“ 1)p (S G n l )p -» B ((SG n_ 1) p .

The fa c t that the following

square B S O (n -l) -------------- - B (S n - 1 ,SGn_ 1)

i

BSO(n) ----------- - --------- - BSG n- i is homotopy ca rte s ia n im plies that

i : (B S O (n -l))£ -» (BSO(n))^ corresponds

to J : (BSO (n))£ -» ( B S G ^ P g . Moreover,

^An_ 1 and \fin fit in a homotopy

comm utative square iff (B S O (n -l » £ -----------

i

(BSO(n))^'

(B S O (n -l))£

i

(BSO(n))g

;

83

9. APPLICATIONS TO TOPOLOGY

b ecau se the horizontal arrows of this square are homotopy e q u iv alen ces, th is square is n e c e s s a rily homotopy c a rte s ia n .

We conclude that and J °ipn for any

i : (B SO (n-l))^' -» (BSO(n))^ corresponds to both J n>0,

so that J = J

in

. ■

We re ca ll th at an 0,-spectrum X and maps

in : 2 X n -> X n+1

a homotopy eq u iv alen ce

is a seq u en ce of pointed s p a c e s

for n > 0 su ch that the adjoint of e a ch

i n is

X n -> ^ X n+1 . The J-homomorphism J : BSO ->

BSG exten d s to a map of (co n n ected ) O -sp ectra Adams operation ^

Xn

J : bSO -» BSG and the

: BSO -» BSO determ ines a map of O -spectra

: (bSO)1 ^ -> (bSO )1 ^ . It is natural to ask whether the maps J , J o ^ q : (bSO)x /q -> (B S G )1 ^q are homotopic as maps of sp e c tra . fortunately, this is not the c a s e :

J , J

homotopic as maps of H -s p a c e s , so that J

: (B S O )1 ^ -> (B S G )^ and J

Un­ are not

cannot have

homotopic restrictio n s to the first delooping of (B S O )1//q ([5 6 ]). N on eth eless, the proof of Theorem 9.1 exten d s to the following spectrum (or “ infinite loop s p a c e ” ) version of the com plex Adams Con­ jectu re proved by the author in [37].

The proof proceeds by carefully

refining S u llivan ’s homotopy th eo retic arguments s o that they remain valid in the much more rigid co n te x t of sp e c tra .

The key step is to interpret a

homotopy c la s s of maps of sp e ctra into (Z /f ^ ^ B S G ) in terms of a geom etric stru ctu re which a ris e s from alg eb raic geometry. T H E O R E M 9 .2 .

L e t J : bU -» BSG

of co n n e c te d £l-spectra and

d en o te the ( co m p lex) J-homomorphism

: (b U ^ ^ ^ (b U )j ^

d en o te the q-th Adam s

operation on the sp ectru m (b U )1 / q for som e q > 0 .

Jo 1 , where ?

n = l0 ,** * ,n },

is the categ o ry of finite pointed s e t s ,

p- : n -> 1 s a tis f ie s

p^(i) = 1 , P j(j) = 0. for j ^ i .

be obtained by ca te g o rica l co n stru ctio n s: BSG a rise from 3~-spaces. J,

and

Such “ ^ - s p a c e s ” can

in p articu lar, both bU and

The proof proceeds by verifying that

J o ¥ q : ( Z /£ ) ooo b U . ( Z / f ) oooBSG

are “ hom otopic” maps of 3"-s p a c e s for any prime Homotopy c l a s s e s of maps of 3"-s p a c e s

B -> ( Z / l ) ^ ° BSG are se e n

to be in natural one-to-one correspon d ence with S2 -fibrations ” over B .

“ Z /£-com pleted

The p re cise definition of su ch a structure

([3 7 ], Definition 7 .2 ) is quite su b tle, having been modified repeatedly to permit such a c la s s ific a tio n theorem for maps into ( Z /f ) ^ ° B S G . The pull-back of the universal via the map J

Z /f-com p leted

S2-fibration over ( Z / l ) ^ ° BSG

is represented by a structure arisin g from algeb raic

geometry (an elab o rate version of holim ° ( Z / £ ) oo °B (A ^ - { o ! ,( G L n £ ) ) e j- -> holim ° ( Z /£ ) oo°B (G L n ^ )et which refines the com plex analogue of (B S O (n -l))^ -» (BSO(n))g in the proof of Theorem 9 .1 ). The map ^

: (Z /Q ^ o b U -> ( Z / Q ^ b U

corresponds to a galois action

of the alg eb raic geometry model (an elab orate version of ae t : holim ° ( Z /E ) ^ ° B ( G L n^C)e t - holim ° ( Z / e ) 0O° B (G L n>c)et )

and is thus covered by a map of

Z /£-com pleted

S 2 -fibrations.

quently, the c la s s ific a tio n theorem implies that J

a nd J

C o n se ­ are

homotopic maps of 3"-s p a c e s into ( Z /f ) ^ ° BSG . ■ D. Quillen once su g g ested that the co n stru ctiv e a s p e c t of algeb raic geometry could provide a valuable approach to various e x is te n c e problems in alg eb raic topology.

Theorem s 9 .3 and 9 .5 (described below) are

exam ples of the s u c c e s s fu l ap plication of th is philosophy.

85

9. APPLICATIONS TO TOPOLOGY

We first con sid er the problem of co n stru ctin g a map between (lo ca liz e d ) c la ssify in g s p a c e s of com p act, con nected L ie groups which is not the cla ssify in g map of a homomorphism. ch aracterized th o se maps B G

In [4], J . F . Adams and Z . Mahmud

> BG which could be “ defined after

finite lo c a liz a tio n ’ ’ in term s of “ a d m issib le ” maps between the u niversal covering s p a c e s of the a s s o c ia te d maximal to ri.

The lo ca liz a tio n n e c e s ­

sary before Adams and Mahmud were assu red of the e x is te n c e of a map on cla ssify in g s p a c e s included the inversion of all primes occurring in the Weyl groups of G and G '.

C onsequently, the following e x is te n c e theorem

provides a sig n ifican t sharpening of one a s p e c t of the work of Adams and Mahmud. One in terestin g a s p e c t of th is theorem is its u se of c h a ra c te r is tic

p

alg eb raic geometry (a s did the proof of the com plex Adams C onjectu re given in [29]).

The theorem its e lf is stated without proof in [36], and the

proof given is an e a sy g en eralization of that given in [33] for a le s s general co n te x t. T H E O R E M 9 .3 .

L e t G(C) an d G '(C )

group s and let f : GF -> Gp

be a homomorphism of a s s o c ia t e d a lg eb ra ic

gro ups over S p e c F , w h ere F F .

he co m p lex re d u c tiv e a lg e b ra ic

is the a lg e b ra ic c lo s u re of the prim e fie ld

T h en f and a c h o ic e of em bedding of the Witt v ecto rs of F

(cf . P roposition 8 .8 ) determ in e a map $ : ( B G ( C ) ) 1 /p - ( B G '( C ) ) 1 /p fitting in a homotopy com m utative sq u a re

B^. is the lifting of the restrictio n of f : Gp -> Gp of GF

to maximal tori

and G p .

P roof (Sketch).

A ch o ice of isomorphisms Tp

^

G L^r^

and

T p ^ G L * r^ determ ines an isomorphism between the group of r x i integer-valued m atrices and the group of alg eb raic group homomorphisms from Tp to

T p . T hu s, we observe that the re strictio n of f ,

: Tp -» T p

fits in a com m utative diagram of group sch em es

tf

TC

Tp-------------

T£,

where R denotes the Witt v e cto rs of F

and where T^, -> TR , T£, -» T p

are determined by the ch osen embedding R -> C . By Prop osition 8 .8 ,

f induces a map

(B G (C ))£ -> (B G ^C ))^

restrictin g to (B (B T '(C ))£ for any holim Su° (

/ .

B ecau se

I ^ p,

where

( )£ =

H *(B G (C ),Q )® Q j = H *((B G (C ))g',Q) and

H *(B G /(C ),Q )® Q g = H * ((B G '(C ))p Q ),

we conclude that (B ( B G ^ C ) ^ ^ finite sk eleta of (B G (C ))1 ^/p .

whose homotopy c la s s is well defined on T o prove the uniqueness (up to homotopy)

of th is map O , we employ the Milnor e x a c t seq uen ce * ^ W lH o m j^ S s k ^ C B G C C )) ! /p ),(B G X C ))1 /p )j ■ ^ { H o n i j , (s k ^ B G C C ))!

We re ca ll that H -((B G (C ))1 is even.

),(B G X C ))1

)!-* .

and 77-|((BG/(C ))1 ^ )

are finite u nless

i

T h erefore, obstruction theory im plies that each of the groups

Homj^ ( 2 s k n(B G (C ))1 ^ , (B G /(C ))1 ^ ) zero .

((BC C C )^ /p ,(B G '(C ))1

is fin ite, so that the

lin^-term is

T his im plies the uniqueness of O . ■

As a corollary of Theorem 9 .3 , we obtain the following e x ce p tio n a l eq u ivalen ces exhibited by the author in [3 3 ], the la s t of which were first exhibited by C . Wilkerson in [71].

T h e se eq u iv alen ces are th o se d eter­

mined by the 11 ex c ep tio n a l is o g e n i e s ” of alg eb raic groups. COROLLARY 9.4. ( B S 0 2n^ 1) 1

a.)

T h e re e x is t s a homotopy e q u iv a len ce

-> (BSpn) 1//2

for any n > 0 .

eq u iv a len ce $ : (B G 2) 1 ^ -> (B G 2) 1 ^

b .)

T h e re e x is t s a s e l f

w hich re s tric ts to B0|. : B T 2 ->BT2 ,

w here ^. s e n d s a short root to a long root an d a long root to 3 tim es a short root,

c .)

T h e re e x is t s a s e l f e q u iv a le n ce $ : ( B F ^ ^ ^ (^ ^ 4) 1/2

w hich re s tric ts to B0^. : B T 4 ->BT4 , w here

s e n d s short roots to long

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

88

roots and long roots to 2 tim es short ro o ts. d u ctiv e L i e group and any prime p > 0 , O : (B G (C ))1 /p - (BGCC))! /p

d .)

For any co m p lex r e ­

there e x is t s a s e l f e q u iv a le n c e

s u c h that 0 * : H2n(B G (C ),Q ) - H2n(B G (C ),Q )

is m ultiplication by pn for e a c h n > 0 . ■ The com parison theorem s of Chapter 8 immediately imply that g alois a ctio n s determ ine s e lf eq u iv alen ces on Z /£-com pletion s (but not n e c e s ­ sarily on ration al cohom ology).

It would be in terestin g to understand how

the se lf eq u iv alen ces of ( Z /£ ) oo°S in g .(B G (C )) determined by G a l(F ,F p ) as in P rop osition 8 .8 depend on the ch o ice of embedding of the Witt v e cto rs into

C.

R ecen t work of Z . Wojktowiak [72] appears to answ er

this question for G(C) = G L n(C ) and I > n . In the following theorem , we present a re la tiv izatio n of Theorem 9 .3 concerning h o m o gen eo u s s p a c e s which w as proved by the author in [36]. THEOREM 9 .5 .

L e t G = G (C ), G ' = G '(C )

g ro u p s; let G ^ , G^ a n d let Spec Z .

Fp,

be a s s o c ia t e d C h ev a lley group s c h e m e s over Spec Z ;

C G^ , H^ C G^ L e t f : Gp -» Gp

groups over S p e c F ,

b e co m p lex red u ctiv e L ie

be c lo s e d su bgro u p s c h e m e s red u ctiv e over

b e a homomorphism of a s s o c ia t e d a lg eb ra ic

w here F

is the a lg eb ra ic c lo s u re of the prime fie ld

s u ch that f re s tric ts to f| ‘ Hp -> Hp .

em bedding of the Witt v ecto rs of F

T h en f ,

f| , and a c h o ic e of

into C naturally d eterm in e a homotopy

c la s s of maps ® :( G /H ) i / p - ( G 7 H ') 1 / p .

P roof (Sk etch ).

The map 0 ^

: ( G / H ) ^ -» ( G '/ H ') ^

is defined as the

unique (up to homotopy—cf. [36]) map fitting in a “ map of fibre tr ip le s ” (G /H )(0 ) --------------- - (B H )( 0 ) --------------- - (B G )(0)

(G '/H ')( 0 ) --------------- -- (B H ')( 0 ) ---------------

(B G O (0)

9. APPLICATIONS TO TOPOLOGY

89

whose middle and right v e rtica l arrows are obtained a s in Theorem 9 .3 . We re call that G /H

is naturally homotopy eq uivalent to B (G ,H ,* ). More­

over, the proof of P rop osition 8 .8 ap plies to prove that each map of the chain (determined by a c h o ice of embedding of the Witt v e cto rs (B (G F ,H F ,* ) e t) ^

R

into C )

(B (G R ,H R , * ) e t) ^ ( B ( G c ,H c ,* ) e t)£' (B (G c ,H c ,* ) s>et)£ -»(Sing. (B (G ,H , *)))'£

is induced by a weak eq u iv alen ce in pro-K^ and is thus a homotopy eq u iv alen ce, where

( )£ = holim S u ° ( / .

The map

(G /H )£ -> (G '/H ')£

is that induced by B (f, th ese eq u iv alen ce s. The com m utativity in

(g /h)(o>—*

of the following squares

n«G/H)p(o) ^ U

o

m -

m>~

( 0) £ £

(G7Ho(0)

U p

p

----- U ^ V io r----£/p

n(Q'/H'), Up

follow s from the uniqueness of O p ( ( G / H ) p ^ -> ( ( G '/ H 'J p ^ “ map of fibre tr ip le s .”

fitting in a

C onsequently, the finite dim ensionality of G/H

implies that th e se sq uares determine a unique homotopy c l a s s of maps V ..

s a tis fy the condition that g be a map of

rigid hypercoverings over f and w hose maps are com m utative sq uares of rigid hypercoverings over f . Define H R (f) whose o b je cts

g : U . . -» V ..

S p (f) to be the full su bcategory of

sa tis fy the condition that U .. -> V .. x y

be sp e c ia l (cf. proof of Prop osition 3 .4 ). Define the homotopy fib re s fib(fj1^_), fib(fe t) , and fib(fgp) by

fib (fht)

= l ( 7 r ( A g ) ' ) - 1(y ) ; g e H R ( f )S

fib (fe t )

= K ^ A g f r ^ y ) ; g e H R R (f )i

fib(fs p ) = i(» U .. -*V.. x y> X . ). We define ^ r':H R (f)

g : U . . -» V ..

to

V .. ,

95

10. COMPARISON OF FIB R E S

s o th at there are ca n o n ica l isomorphisms ^

n (A r(g ))~

in H J

and ( F ( A ^ ( g ) ) 'r 1(y ) ^

( tK A ^ X b ) ) ' ) - 1 ^ )

^

H* .

fib(f^t) via the functor x\s and the natural maps

We define fib(fg p)

MA^g))*)-1^) ^ (rr(Ar(g))'r\y) — (r7-(Ag)~r 1(y). T he reader can e a s ily verify that fib(fg p) -» fib(fj1^.) is homotopy inverse to fib(fht) "* fib(fsp ) nsing the fa ct that

g -> ^r(g) -♦ (9(g) and g -> \Jj '(g ) -» 0(g)

are homotopic. T o verify the secon d a sse rtio n , we observe that the com position (X . x Y> y )ht -> fib(fht)

fib( fsp)

is given by the functor i* ° s : Sp(f ) -*H R (X . x y > y ,y )

sending

g : U . . -» V ..

to

U .. x y y . The proof of the a ss e rtio n follow s

from the e x is te n c e of natural in clu sion s HR(X. x y y , y)

U .. x v

v -* U .. x y y in

determ ining a natural transform ation

rj -> i * ° s . ■

The next proposition exp lain s our introduction of Sp(f ) and our co n ­ sid eration of fib(f s p ) .

The proposition originally appeared as P rop osition

2 .4 of [29]. PROPOSITION 1 0 .4 .

L e t f : X -» Y

s c h e m e s and let fjTp(y) £ Pro'^ *

fsp1^ ) = ^77(u - x v.

b e a p o in ted map of lo ca lly n o etherian

b e d e fin e d by

g : U - ^ v - in s P(f )! = S ^ Cg r Hy ) ; g f S p ( f ) i

96

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

(w here vQ -» V.

is the in clu sio n of the d is tin g u is h e d com ponent of VQ ).

For any locally co n sta n t a b elia n s h e a f M on E t ( X ) ,

th ere e x is t s a

natural isom orphism

H*(f s p ( y ) ’ M)

(R *f*M )y

w hich fits in a com m utative sq u a re

^ (R*f*M)y

H *((X y )h t,i*M ) -------—

-------

H *(X y ,i*M )

w here i : X y = X x y y h>X is the g eo m etric fib re of f ,

the left arrow is

in d u ced by the map (X y )^ -> f^"p(y) g iv e n by rj of P roposition 1 0 .3 , the right arrow is the ca n o n ica l b a s e ch a n g e map, a n d the bottom arrow is the isom orphism of P roposition 5 .9 . P roof. The functor

77: Sp(f )

HR(Xy,y) of P rop osition 1 0 .3 induces the

following map of s p e c tra l se q u e n ce s (whose co lim its are indexed by g : U.

V.

in S p (f) and W.

in HR(Xy,y) ) b e ca u s e

Up x y

vQ -» X x y vQ

is a hypercovering

E j '^ = colim H^(Up x v

'E ^ ’q =

vQ,M) = > colim HP+9(U . x y vQ,M)

colim Hq(Wp,i*M )

=> colim HP+C1(W. ,i*M ) .

By P rop osition 3 .7 , colim H*(U. x y vQ,M) = colim H *(X x y vQ,M) = (R*f^M)y and colim H*(W. ,i*M ) = H *(X y ,i*M ) .

10. COMPARISON OF FIB R E S

97

X y -» X x y vQ

The map on abutments is induced by the clo se d immersions and is therefore the b a se change map.

As se e n in Theorem 3 .8 , the

sp e c tra l seq u en ce c o lla p s e s at the E 2 -le v e l with E * > ° ^ H * ( ( X y )h t,i*M )

a s in P rop osition 5 .9 .

Sim ilarly, the

E^^

s p e c tra l se q u e n ce s c o lla p s e s

at the E 2-lev el (use the proof of Theorem 3 .8 with the presheaf G3 rep laced by H^( ,M) ) and E 2 ' ° = colim H*(77(U. xv v ),M )). L e t H *(fSp1(y),M )

(R*f^M)y be the edge isomorphism

E *’0

E ^ . Then

the a sse rte d com m utative diagram is a co n seq u en ce of the naturality of the edge homomorphism. ■ Although P rop osition 1 0 .4 has been presented only for maps of sch em es, its ap p licab ility to maps of sim p licial sch em es is a con sequ en ce of the following lemma. L E M M A 1 0 .5 .

L e t f : X . -> Y .

b e a po in ted map of lo ca lly n o etherian

sim p licia l s c h e m e s with the property that e a c h X R a nd ea c h Yn is c o n ­ n e c t e d , and let M b e a lo ca lly co nstant a b elia n s h e a f on E t ( X .) . map (X . Xy y \ t -» fib(f ea ch

If the

) of P roposition 1 0 .3 in d u ces isom orphism s for

n > 0 H *(fib ((fn)s p ),Mn) ^

H *((X n x y

y )h t,i*M n) ,

then this map a lso in d u c es an isom orphism H *(fib(fsp ),M) ^

Proof. F o r any g : U . . -»V ..

H *((X . xy> y )h t,i*M ) .

in H R (f),

defined by the condition that ( 77(g) ) for each

n > 0.

Then

A ( 77(g)~) -» 77(A g )~ : sim p licial s e ts

n(Ag)

let 77(g) :77(U ..)

tt^ V ..)

be

be the mapping fibration of ^(gn)

77(Ag)~ fa cto rs through a natural map

this is proved by observing for any map of bi-

h :S .. -> T ..

and any

: A[n] -> A[n] (the inclusion of

s k J1_ 1A[n] minus the i-th fa c e into A[n] ) th at a map e-

hn

naturally

98

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

determ ines a map a^ -» A (h ). Moreover, b e ca u s e con nected for each

n > 0,

77(Un ) and 7r(Vn ) are

Theorem B .4 of [12] im plies that

A w g y r ^ y ) - > ( ^ ( A g ) 'r 1(y ) is a weak eq u iv alen ce. C onsequently, the natural map f Z

is a pointed

fa cto rs through g ,

then

7 0 s in c e

colim H^(v x z Z ',M ) = 0

for

q > 0

y->v

and any geom etric point y of Z rep laced by H^(

(cf. proof of Theorem 3 .8 with G3

x z Z ',M ) ). ■

We now present our b a sic com parison theorem , a g en eralizatio n to sim p licial sch em es of [2 9 ], Theorem 4 .5 . THEOREM 1 0 .7 . su ch that ea ch ea ch

77'1((Yn)j1^;)

L e t f : X . -* Y .

b e a p o in ted map of sim p licia l s c h e m e s

X n is c o n n e c te d , ea c h Yn is c o n n e c te d and no eth eria n , is p ro fin ite, a n d n 0(X . x y y )

lo cally co n sta n t a b elia n s h e a f on E t(X .)

is fin ite . L e t M be a

s u c h that for e a ch

n,q > 0 ,

R (fn)” i ( y ) ”

(R qfn*Mn)v H*f(Xn x y y,i*M n) implies that y n induces an isomorphism

H *((fn)sJ(y)> Mn) ^ by P rop osition 1 0 .4 .

H* « X n > 0 *

We let C denote the left d irected categ o ry of co n n ected , pointed, p rin ci­ pal G -fibrations of fn by Y n

Y ^ n> Y n , and we le t f ^ : Yn • T he maPs

(f n)Sp

-> X n denote the pull-back

(fn)sp

indexed by

-» Yn in C

determ ine maps of sp e ctra l seq u e n ce s inducing isomorphisms on abutments

E 2*q(Yn

Yn) =

HPCCY^t.H'kfibCf^.M)) => H ^ X ^ M )

' E P ^ Y ; ^ Yn) = H P((Yn)h t, H'l((f n)3 J(y ),M )) = > HP+cl(X n,M) . The hypothesis that R ^ n5ie'Mn is lo ca lly co n sta n t on Et(Y n) im plies that R ^ f^ M n (the re strictio n of the fa ct th at

) is lo cally co n stan t on E t ( Y ^ ) ;

' e P, c* ca n be identified with the L eray s p e ctra l seq uen ce

im plies that the c o e fficie n ts for the cohomology groups of (Y ^ )j^ lo ca l co efficien t system in the c a s e of

are a

^ a s we^ as *n the c a s e °f

. (We have im plicitly used the noetherian hypothesis on Yn to obtain the E 2-term s in the above form by taking a d irect limit of c o e ffi­ c ie n ts ; for more d e ta ils , s e e [29], 4 .3 .) The map f^ -> fn cle a rly induces an isomorphism on geom etric fibres x n xY; y ' ^ n

x n xY y • n

M oreover, Lemma 1 0 .6 im plies that fib(f^) -» fib(fn) is an isomorphism in p ro -K ^ . We tak e the colim it (indexed by C °P ) of the maps

E P ,q (Y ; - Y n) s o that colim

' E P ' ^ y ; - Y n)

E ^ ’^ is cohomology with triv ial c o e fficie n ts (the a ctio n

of „ (Yn) on R Yn ); moreover, colim E V *

is a ls o cohomology with triv ial co e fficie n ts b e ca u s e

77i((Y n)j1j:)

is pro-finite and the a ctio n of 77i((Y n)]l t ) on H *(fib(fn),M) is continuous.

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

102

Comparing th e se colim it sp e ctra l se q u e n ce s, we conclude that the map H *(fib ((fn)s p ),M) ^

colim H * ( f i b ( ( f ^ p),M)

— colim H * ( (Q -l (y ),M ) ^

H *((fn)-J(y ),M )

is an isomorphism a s required. The proper smooth b a se ch ange theorem ([7 ], X V I.2 .2 ) a s s e rts that f : X -» Y

and M e AbSh(X) a s below s a tis fy the h ypotheses of Theorem 1 0 .7 .

C O R O L L A R Y 1 0 .8 .

L e t f : X -> Y

b e a p ro p er, sm ooth, po in ted map of

c o n n ected , no eth eria n s c h e m e s and let M be a lo ca lly co n sta n t, c o n ­ stru ctib le s h e a f on E t(X )

with sta lk s of order rela tiv ely prim e to the

re s id u e c h a ra c te ris tic s of Y .

If ^ 0 ^ )

is pro-finite, then the natural

map (Xy >et -> fib(fe t ) in d u c e s an isom orphism H *(fib(fe t ),M) ^

H *((X y )e t ,M) . ■

In the proof of the Adams C onjectu re presented in [2 9 ], the following s p e cia l c a s e of Theorem 1 0 .7 was required (cf. [2 9 ], 5 .3 ). proof that fib(f^t) is weakly

An independent

£-equivalent to a sphere has been pro­

vided by D. C ox in [21] (using the e x is te n c e of a “ Thom isom orphism ” ). The fa ct that the map f : X -> Y

and the ab elian group of C orollary 1 0 .9

sa tisfy the hypotheses of Theorem 1 0 .7 is verified in P rop osition 5 .2 of [29]. C O R O L L A R Y 1 0 .9 .

L et Y

b e a pointed, c o n n e c te d , g eo m etrica lly u ni­

b ra n ch ed n oetheria n s c h e m e and let E on Y .

L e t V (E ) = Sym (Ev)

be a co h eren t, lo ca lly fr e e s h e a f

be d e fin e d lo ca lly as the sp ectru m of the

sym m etric a lgeb ra of the dual of E

over Oy

and let f : X -> Y

the stru ctu re map V ( E ) - o ( Y ) -> Y , w here o : Y -» V (E ) For any fin ite a b elia n group A c h a ra c te ris tic s of Y ,

d en o te

is the 0 -se c tio n .

of order rela tiv ely prim e to the re s id u e

the natural map (X y )et -> fib(fe t)

isom orphism in cohom ology H *(fib(fe t),A ) ^

H *((X y )e t ,A ) . ■

in d u ce s an

11.

A PPLICA TIONS TO GEOMETRY

In th is ch ap ter, we present four ap p licatio n s of e ta le homotopy theory to geom etry.

Theorem 1 1 .1 is a resu lt of P . Deligne and D. Sullivan

a sse rtin g that a “ f l a t ” v ecto r bundle ca n be triv ialized by p assin g to a finite covering s p a c e of the b a se .

T he next ap p licatio n , C orollary 1 1 .3 ,

is a n e ce ssa ry and su fficien t condition due to M. Artin and J . -L . Verdier for a real alg eb raic variety to have a real point; the proof we present is based on a resu lt of D. Cox concerning the e ta le homotopy type of a real v ariety.

Theorem 1 1 ,5 p resen ts a long e x a c t homotopy se q u en ce a s s o c i ­

ated to certain maps ( “ geom etric fib ratio n s” ) of sch em es and their geom etric fib res.

F in a lly , we verify in Theorem 1 1 .7 that a smooth a lg e ­

braic variety over an alg e b ra ica lly clo se d field

k has a b ase of e ta le

neighborhoods w hose e ta le homotopy types have 2-com pletions which are K(77,1 ) ?s

for £ a prime invertible in k .

We re c a ll that a com plex v ecto r bundle over a pointed, connected b ase has a d iscre te stru ctu re group if and only if it a ris e s from a com plex rep resen tation of the fundamental group of the b a se (a s the bundle a s s o c i ­ ated to the corresponding lo ca l co e fficie n t sy stem whose fibres are com ­ plex v ecto r s p a c e s ).

If the b a se of a finite dim ensional, com plex v ecto r

bundle is a manifold, then the bundle has d isc re te stru ctu re group if and only if the bundle adm its an integrable con nection (in which c a s e , the bundle is said to be “ f/a f” ).

The co n clu sio n of the following theorem

describ in g how su ch a v e cto r bundle can be triv ia liz e d (but not ad dressin g the q uestion of triv ializin g the a s s o c ia te d rep resen tation of the funda­ mental group of the b a s e ) is a homotopy th e o re tic statem en t amenable to e ta le homotopy th eo retic tech n iq u es.

103

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

104

T h e o r e m 1 1 .1 (Deligne-Sullivan [25]). com plex and let p : E -> T

b e a fin ite sim p licia l

b e a fin ite d im en sio n a l, co m p lex v ecto r bundle

with d is c r e te stru ctu re gro u p . sp a ce T '-> T

L et T

T h en th ere is a fin ite, s u rje c tiv e co v erin g

s u c h that the p u ll-b a ck p ' : E ' - > T ' of p to T '

is a

trivial v ecto r b u n d le. P roof (S k etch ). con n ected .

We are e a sily reduced to the c a s e

is pointed and

p : 7r1(T ,t) -> G L n(C ) be a rep resen tation determining

Let

p : E -» T . B e c a u s e

77'1(T ,t)

is finitely gen erated , we may find a subring Z su ch th at p is given by p : zr1(T ,t) ->

A of C of finite type over G L n(A ).

T

D eligne and Sullivan show that the c la s s ific a tio n map for

p , f : T -> B U n , ca n be represented up to homotopy by a map g A( C ) : X A(C )t0 P - G ra s s N+n n . A(C )to P , where

g A : X A '* G r a s s N + n ,n ;A

is a map of sch em es defined over A . Let

m and

m' be maximal id eals of A whose (fin ite ) residue fields

A /m = k and A /m ' = k / have d istin ct c h a r a c te r is tic s .

An argument

sim ilar to th at of the proof of P rop osition 8 .8 v erifies that

gA has the

property th at (g A(C )to P /

(g p )^ . , where

L

is tic of k (re s p ., K 01

Let

T '^ T

(g ^

and (g A(C )to P /

~

L ' ) denotes the s e t of a ll primes e x ce p t the c h a ra c te r­

(re sp e ctiv e ly ,

triv ial if (g r )

~

k ' ).

T his im plies that (g A(C))^°P is hom otopically

and {g-j-,) K

0L

are hom otopically triv ia l.

be the fin ite, s u rje ctiv e , pointed covering sp a c e a s s o c i ­

ated to the subgroup (ker p^) fl (ker p ^) of zr1( T , t ) , where p 1 : tt^{T ,t)-^ G L n(k) and p [ : 7r1(T ,t) -> G L n(k r) are induced by p : 7r1( T ,t ) ^ G L n( A ) . Applying the above argument to p ' : ^ ( T ^ t ) -> G Ln( A ) , we conclude that the cla ssify in g map for p ' : E ' - > T ' is hom otopically triv ial. ■ The following result of D. Cox d escrib e s the etale homotopy type of a real algebraic variety in terms of its a s s o c ia t e d complex analytic variety.

11. APPLICATIONS TO GEOMETRY

Let X

PROPOSITION 1 1.2 (D. Cox [20]).

105

b e a c o n n e c te d re a l a lg e b ra ic

variety ( i . e . , a r e d u c e d , irred u cib le s c h e m e of fin ite type over Spec R ), and let X ^ P

d en o te the co m p lex analytic s p a c e of co m p lex points of X .

T h en th ere is a w eak homotopy e q u iv a le n c e of Artin-M azur co m p letio n s (w ith r e s p e c t to the s e t P

of a ll p rim es)

x et

w here G is the group

P roof.

Let

group G . and

Z /2

~ (X j?P x |EG|)P

a ctin g on

by com plex conjugation.

Xp = X x Spec C , so that X p -» X L Spec R L Let

U. = c o s k ? ( X p ) ; L

b ecau se

U. =* X

is g alo is with

x cosk^Pe c ^ (S p ec C) Spec R

770(coskQPe c ^ (S p e c C )) = BG , there is a natural homeomorphism |ut°p|

as

|EG| .

X

^

G

The proposition now follow s d irectly from P ro p o sitio n 8.1 and Theorem 8 .4 . ■ The following criterion w as first proved by M. Artin and J . -L . Verdier and published by D. Cox with a proof based on P rop osition 1 1 .2 .

We

present another proof based on th is proposition. C o r o l l a r y 1 1 .3 (D. Cox [20]).

L et X

variety of dim ensio n n .

has at le a s t one rea l point if and only

T h en X

b e a c o n n e c te d rea l a lg e b ra ic

if H*(Xe j.,Z /2 ) A 0 for som e i > 2n . P roof. map X

If X

has a real point, then th is point is a se c tio n for the structure

Spec R .

B ecause

i > 0 , we conclude that

H ^.(Spec R, Z /2 ) = H * ( B Z /2 ,Z /2 ) ^ 0 for any

H *t( X ,Z / 2 ) ^ 0 for any i > 2n w henever X

a real point. C o n v ersely , if X

has no real point, then G = Z /2

has

a c ts

freely on X 4° P , s o th at X *°P x |EG| is homotopy equivalent to X ^ ?P /G . t b Q L B e c a u s e X^?P/G is a com plex an a ly tic v ariety of com plex dimension n and

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

106

Hi (X ^ ? P /G ,Z /2 )

^ H \ X e V Z /2 ) by P rop osition 11 .2 , we con clu d e that

H1(X e ^ Z / 2 ) = 0 for all

i > 2n whenever X

has no real points. ■

In [48], C orollary X .1 .4 , A. Grothendieck proved the e x a c tn e s s of the following seq u en ce 771(X y ) - 7 r 1( X ) - 7 7 1( Y ) - l whenever f : X -> Y

is a proper, smooth pointed map of noetherian, normal

sch em es with con n ected geom etric fibre X y . T h is seq u en ce was g en eral­ ized by M. Raynaud based on n otes of Grothendieck ([4 8 ], X III.2 .9 and X III.4 .1 ) to the asse rtio n that the following seq u en ce is e x a c t ^ ( X y ) 1- - ffj(X )/N - ^ ( Y ) - 1

where N is the normal subgroup of ^ ( X ) N = ker(ker(771(X )

L

defined by

^ ( Y ) ) -> k e r ^ X )

771( Y ) ) L ) ,

is the s e t of primes com plem entary to the residu e c h a ra c te r is tic s of Y ,

and f : X -> Y

is the re strictio n of f : X -* Y a s above to X = X - D

D is a d ivisor in X

with normal c ro s s in g s over Y

with

(a s defined below ).

We proceed to extend this la tte r homotopy seq u en ce to higher homotopy groups, observing that L-com p letion is e s s e n tia l (for exam ple, the Kunneth Theorem fa ils com pletely for c o e fficie n ts not prime to residu e c h a ra c te r­ i s t ic s ).

The homotopy seq u en ce will be proved for “ geom etric fib ratio n s”

which we now define. DEFINITION 1 1 .4 .

A map f : X -> Y of sch e m es is said to be a sp e cia l

geom etric fibration if f is the re strictio n of a proper, smooth map f : X ->Y to X - T ,

where T

condition:

T

c- in X 1

is a clo se d subschem e of X

is the union of clo se d su bschem es

over Y

is smooth over Y

sa tisfy in g the following T- of pure codim ension

su ch that e a ch non-empty in te rse ctio n of pure codim ension

c-

T fl-flT j A1 s

+••• + c- . (If e ach xs

c- = 1 ,

107

11. APPLICATIONS TO GEOMETRY

then T

is said to be a d ivisor in X

More g en erally ,

f:X

ea ch

V j, f j : f _ 1 (Vj)

is said to be a g eo m etric fibration if Y

Y

a Z arisk i open covering

with normal c ro s s in g s over Y .) admits

{Vj->Yi su ch that the re strictio n of f above

V j,

is a s p e c ia l geom etric fibration.

A geom etric

fibration (of re la tiv e dim ension 1 ) is said to be an elem en ta ry fibration if its geom etric fibres are co n n ected , affine cu rv e s. ■ In the following theorem (Theorem 4 .2 of [3 1 ]), our con sid eration of lf(i) : X (i) -> Y(i)S is a formal means of p assin g to the ( Y (i)

of Y , and let

b e the p u ll-ba ck of f by Y (i) - Y .

T h en

Y

there e x is t s a long e x a c t homotopy s e q u e n c e

w here L

- ” n « X y )e t) -

" V !Y « e t *> - -

is the s e t of prim es com plem entary to the re s id u e c h a ra c te ris tic s

of Y . P roof (S k etch ).

By Lemma 1 0 .6 , the natural map fib (!f(i)e ^J) -» fib(fe t) is

a weak eq u ivalen ce. map fib (jf(i)e t !) L

B ecau se

{Y (i)

is simply co n n ected , the natural

fib ({f(i)^ .j) is a weak eq u iv alen ce.

T hu s, it su ffices

to prove the natural map (X y)e t -> fib(fe t) is a weak L -e q u iv a le n ce ( i .e ., s a tis f ie s conditions a) and b) of C orollary 6 .5 ). Let

K denote

ker(77‘1(X e t ) -> 7r1(Ye t) ) L . B e c a u s e the rem oval of

smooth clo sed su bschem es of pure codim ension g reater than 1 d oes not affe ct the fundamental group ([7 ], XVI. 3 .3 ), we may apply the above Raynaud e x a c t se q u en ce to con clu d e that

771((X y )^ .) ->K is su rje ctiv e

108

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

with kernel H .

One v erifies that H is cen tral in 771((X y )^ j.). The

homotopy seq u en ce im plies that H ',

the kernel of the su rje ctiv e map

77’1(fib(fe j.)L ) ->K is a ls o ce n tra l. L e t M be a lo ca lly co n sta n t, co n stru ctib le , L -to rsio n abelian sh eaf on X (i) w hose re strictio n to some L-prim ary g alo is exten sio n is co n stan t.

X (i) '-> X (i)

We verify below th at we may apply Theorem 1 0 .7 to

f(i) and

M , concluding that (X )e j. -> fib(fe t) induces isom orphism s ( 1 1 .5 .1 )

H *(fib (fe t ),N ) _

for any lo cal system

N of finite, L-prim ary ab elian groups on fib(fe ^)

induced from a rep resen tation of K . 1 0 .7 for f(i) and M,

H *((X y )e t ,N)

To verify the hypotheses of Theorem

we may assum e f(i) is a s p e c ia l geom etric fibra­

tion with re la tiv e co m p actificatio n

j : X (i) -* X ( i ) .

A s in the proof of the

proper, smooth b a se change theorem ([7 ], XVI. 2 .2 ), it su ffice s to prove that R ^f(i) M comm utes with arbitrary b ase change on Y (i)

for a ll q > 0 .

U sing the L era y s p e c tra l seq u en ce and the proper b ase change theorem for f ( i ) ,

we are reduced to proving that R^j^M commutes with arbitrary

b ase change on Y (i) for a ll

p > 0

in order to verify the h ypotheses of

Theorem 1 0 .7 for f(i) and M.

T his is verified by exam ining the stalk s of

RPj^M at a geom etric point x

of X ( i ) - X ( i ) , computed a s the limit of

HP(

,M) applied to deleted e ta le neighborhoods of x .

e ffe ct upon (R^j*M )

F in a lly , the

of arbitrary b a se ch ange on Y (i) is controlled

using the coh om ological purity theorem ([5 9 ], V I.5 .1 ) thanks to the follow ­ ing co n seq u en ce of Abhyankar’s Lemma ([4 8 ], X II.5 .5 ):

on a c y c lic

covering of a deleted neighborhood of x , M can be extended to a lo cally co n stan t sh eaf on som e sm ooth, re la tiv e (to Y (i) ) co m p actificatio n . L e t ((X y )^t )H - (X y )L s p a c e s a s s o c ia te d to

( 1 1 .5 .2 )

and (fib (fe t) L ) H^ f i b ( f et) L be the covering

H and

H'.

Then ( 1 1 .5 .1 ) im plies that

« X y )e t)H - W

e P V

11. APPLICATIONS TO GEOMETRY

induces an isomorphism in

Z /£ cohomology for e a ch

isomorphism of (ab elian ) fundamental groups. (and, thus,

109

le L

and thus an

The proof that (1 1 .5 .2 )

(X y )^ . -> fib(fe j.)L ) is a weak eq u iv alen ce is com pleted by

proving that

H

H ' a c ts triv ially on the

Z /2 cohomology of the universal

covering s p a c e s of (Xy)^j. and fib(fe t) L . T h is la s t statem en t is proved using the following to p o lo g ical resu lt ([3 1 ], Appendix):

if F -> E -> B

a fibre triple of co n n ected , pointed s p a c e s , then the actio n of the homology of the universal covering of F

is on

fa cto rs through an actio n of

77^ £ ) . ■ The geom etric b a sis for our la s t ap p licatio n , Theorem 1 1 .7 , is the following theorem of M. Artin. T H E O R E M 1 1 .6 (M. Artin [7], X I. 3 .3 ).

L et X

be a smooth a lg e b ra ic

variety of dim en sio n n over an a lg eb ra ic a lly c lo s e d fie ld k an d let x be a c lo s e d point of X . U of x

T h en th ere e x is t s a Z a ris k i open n eigh bo rh o o d

w hose stru ctu re map U ^ Spec k may b e fa cto red a s a co m p o si­

tion of elem enta ry fibrations : f U - Un

ft > Un_ 1 ------> ••• ------> U 1 ------> S p eck . ■

In the sp e c ia l c a s e in which

k equals

C , Theorem 1 1 .6 a s s e r ts

that X to P admits the stru ctu re of the to tal s p a c e of an iterated fibration with fibres which are co n n e cte d , noncom pact Riemann s u rfa c e s . B e c a u s e the latter are

K(7t ,1 ) , s with n a free group, the homotopy seq u en ce im­

p lies that X to P is a ls o a

K (77,l) with n a s u c c e s s iv e exten sio n of

free groups. T o con clu d e a sim ilar resu lt for p o sitiv e c h a ra c te ris tic v a rie tie s , we must con sid er homotopy typ es com pleted away from the c h a ra c te r is tic of k . T he n o n -exactn ess of com pletion requires us to co n sid er e ta le neighbor­ hoods and only one prime at a time.

The proof we provide for Theorem 1 0 .7

is somewhat sim pler than the original one given in [30].

110

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

T H E O R E M 1 1 .7 .

Let X

c a lly c lo s e d fie ld k ,

let x

prim e in v ertib le in k . in X

be a smooth a lg e b ra ic variety over an a lg e b ra i­ b e a c lo s e d point of X , and let I be a

T h en th ere e x is t s an eta le n eigh bo rh o o d V

su ch that ttj(Ve j.)

of x

is a s u c c e s s i v e e x ten sio n of fin itely g e n e ra te d ,

fr e e pro-£ gro ups and s u c h that the natural map

V

^ K( ^ ( V e t / , l )

is a w eak e q u iv a le n c e . P roof.

The proof of Theorem 1 1 .6 provided by M. Artin in [7] proceeds by

verifying the e x is te n c e of a Z arisk i open Un of X elem entary fibration f n : Un -> therefore smooth). that R 1f ^ Z / £

Let

g:

is open in

(and

, where f ^ :

-» U ^_i

by g . If we ite ra te th is procedure (n ext co n sid e r­

ing the smooth sch em e V of x

P n_1

Un l be a fin ite, e ta le map ch osen so

=* g * R 1fn^ Z /£ is co n stan t on

is the pull-back of f

borhood

where

together with an

in p lace of X ), we obtain an e ta le neigh­

in X w hose stru ctu re map fa cto rs as a s u c c e s s io n of

elem entary fibrations

V = Vn - ^ V

with the property that R 1g -;+cZ /£ Let

-----------> Vj — —» Spec k

l —

is co n sta n t on V-_^

for e ach

i.

denote the geom etric fibre of g^. As argued in the proof of

Theorem 1 1 .5 , we may apply Theorem 1 0 .7 to conclude that (C^)et -> fib((g^)e j.) induces an isomorphism in

Z /2 cohom ology.

B ecau se

C-

is

con n ected and has £-cohom ological dimension 1 , the actio n of 771((V^_1) e ^) on H *((C j)e t ,Z /£ ) co n stan t).

“ H ^ fib C gP gj.Z /E ) is trivial (s in c e

is

We may therefore com pare the Serre s p e c tra l seq uen ce for (g j)e f-

f and (g j)et to con clu d e that fib ((g i) e t) phism in

I ^ g ^ Z /E

I fib ((g i) e t) induces an isom or­

Z /£ cohom ology.

We conclude that the com position (C i)et - f i b ( ( g i)e t) - f i b ( ( g i4 )

11. APPLICATIONS TO GEOMETRY

induces an isomorphism in len ce

(Cp)^.

Z /£ cohomology and therefore a weak eq uiva­

fib((g^)^t) of £-nilpotent pro-sim p licial s e ts . B e c a u s e I (C p ^ is weakly equivalent

is a smooth, co n n ected , affine cu rve over k , to

111

K (77,l) where

77 is a finitely gen erated , free pro-£ group. The

theorem now follow s im mediately by applying the long e x a c t homotopy seq u en ce to the s u c c e s s iv e fibering

12.

APPLICA T IO N S TO F IN IT E CHE V A L L E Y GROUPS

In this ch ap ter, we employ e ta le homotopy theory to determ ine cohomology groups of finite C hevalley groups and homotopy typ es of a s s o c ia te d K-theory s p a c e s . group of Fq-ration al points

In p articu lar, our techniques apply to the G(F^) of an alg eb raic group

over a field

k of finite c h a ra c te r is tic and to related tw isted groups (for exam ple, to

Un(Fq) = l(« ij) € G L n(F q ), ( a y ) , ( a j ) 1 = I„} ). The b a sic theorem of th is chapter is Theorem 1 2 .2 which provides an e ta le homotopy th e o re tic interpretation of an isomorphism of S. L an g. Although Theorem 1 2 .2 was first proved by the author in [34], the re le ­ van ce of L a n g ’s isomorphism w as first observed by D. Quillen in [62]; Quillen a ls o su gg ested formulating the L an g isomorphism a s a homotopy c a rte s ia n sq uare.

C orollary 1 2 .3 re la te s a sta b ility resu lt for the co h o ­

mology of finite c l a s s i c a l groups, w hereas C orollary 1 2 .4 provides a com parison of the cohom ology of d iscre te C hevalley groups to that of the cla ssify in g s p a c e s of the a s s o c ia te d L ie groups.

P rop osition 1 2 .5 d e te r­

mines certain (u n stab le) s p a c e s obtained using Q u illen ’s plus co n stru ctio n . The most in terestin g ap p licatio n of Theorem 1 2 .2 is given in Theorem 1 2 .7 , in which the various unitary K-theory s p a c e s for finite field s are identified. As the reader can a s c e rta in by comparing the m aterial of th is ch apter to the author’s various papers on th e se to p ic s , our p resen tation here is som e­ what sim pler and more d irect than that in the literatu re. The following proposition is a g en eralizatio n due to R . Steinberg [68] of a theorem of S. L an g [5 3 ] to tw isted C h ev a lley groups G ^(k)^ .

112

113

12. APPLICATIONS TO FIN ITE C H E V A L L EY GROUPS

P R O P O S I T I O N 1 2 .1 .

L e t Gk b e a c o n n e c te d lin ea r a lg e b ra ic group over

an a lg eb ra ica lly c lo s e d fie ld and let 0 : Gk -» Gk be a s u rje c tiv e endom orphism su ch that the group of k-rational points of Gk invariant under , H = G k(k ) ^ ,

is fin ite.

T h en the “ L a n g map” 1 / 0 : Gk -» Gk

is a p rincipal H -fibration, w here 1 / 0 g * 0 ( g ) 1 • C o n se q u e n tly , 1 / 0 l/

0

s e n d s a k-rational point g

to

in d u c e s an isom orphism : G k/ H ^ > G k . ■

T h is proposition is particularly striking when one re c a lls that top ologi­ c a l groups have no non-abelian con nected covering s p a c e s .

T hu s, P ro p o si­

tion 8 .8 im plies th at whenever H (a s in P rop osition 1 2 .1 ) is non-abelian and G red u ctiv e, the order of H must be d iv isib le by the residue c h a ra c ­ te r is tic of k . The following theorem (Theorem 2 .9 of [3 4 ]) is our e ta le homotopy th eoretic in terpretation of P rop osition 1 2 .1 .

The square ( 1 2 .2 .1 ) has been

referred to a s the “ co h o m o lo gica l L a n g fib re s q u a r e d 1 T H E O R E M 1 2 .2 .

L e t G(C)

be a com p lex red u c tiv e L ie group, let G ^

b e an a s s o c ia t e d C h ev a lle y in tegra l group s c h e m e , let k be an a lg e b ra i­ ca lly c lo s e d fie ld of c h a ra c te ris tic p , endom orphism with H = Gk(k )^

and let 0 : Gk -> Gk be a s u rje c tiv e

fin ite.

T h en a c h o ic e of em bedding of the

Witt v ecto rs into C d eterm in es a com m utative sq u a re in the homotopy ca tego ry for any prim e I with p | I : B H -----------------------------

( 1 2 .2 .1 )

(Z /O * , ° S in g.(B G (C ))

D

o S in g.(B G (C ))

(Z /Q ^ ° S in g .(B G (C )x 2 )

114

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

with the property that a c h o ic e of map on homotopy fib re s fib(D) -> fib(A) d eterm in ed by { 1 2 .2 .1 ) in d u c e s isom orphism s in Z /£ A

cohom ology, w here

is in d u ced by the diagonal G -» Gx2 .

P roof.

We interpret P rop osition 1 2 .1 as a sse rtin g that the following square

of sim p licial sch em es B (G £ 2 /A (G k),G £ 2 ,* )

B (G k/H ,G k,* )

( 12 .2 .2 )

d

S

1 X(f)

BGk

is ca rte s ia n , with the isomorphism on fibres by the L ang isomorphism

Gk/H

Gk/H

Gk2 /A (G k) given

Gk . The hypotheses of Theorem 1 0 .7

are sa tisfie d with M = Z / £ , b e ca u s e for e a ch sim p licial degree

n both

dn and Hi (SOn(F q) , Z / 0

i < n -2

H^(Spinn+1(F q ), Z /£ ) - .H ^ S p i n ^ F ^ Z / e )

i < n -2 . ■

Another ea sy co n seq u en ce of Theorem 1 2 .2 is our next co ro llary which show s that B G (F )

is a good cohom ological model for B G (C ). Corollary

116

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

1 2 .3 in clu d es P rop osition 1 .3 of [32] (all of the c l a s s i c a l ty p e s) and P rop osition 5 .3 of [52] (e a ch of the e x cep tio n al typ es with £ not dividing the order of the Weyl group). C O R O L L A R Y 1 2 .4 .

A s su m e the notation of T h eo rem 1 2 .2 and let F = F p ,

the a lg eb ra ic c lo s u re of Fp . A s s u m e that G is eith e r of c la s s ic a l type or that H * (T (C ),Z /£ ) -> H *(G (C ),Z /£ ) is s u rje c tiv e , w here T

is a maximal torus of G .

T h e d irect limit ( with

r e s p e c t to q = p^ ) of the maps Dq : BG (Fq) - (Z /Q ^ o Sin g.(B G (C )) a s s o c ia t e d to the fro b en iu s maps cf>^ : Gk -> Gk in d u ce s isom orphism s in hom ology a nd cohom ology H ^ B G C F X Z /Q P roof.

H * (B G (C ),Z /£ ), H *(B G (C ),Z /£ )

C learly , it su ffice s to con sid er homology.

from the p rojection dq : B (G k/G (F q ),G k,* )

H * (B G (F ),Z /£ ).

The map Dq a ris e s

B G k , so that it su ffice s to

show that th e se maps induce an isomorphism colim H *(B (G k/G (F q),G k ,* ) ,Z /£ )

H *(BG k,Z /£ ) .

Using the colim it of the Serre sp e c tra l se q u e n ce s for e ach (dq)e t , we conclude that it su ffice s to prove that colim H^(Gk/G (F q) ,Z /£ ) = 0 . To determ ine the map in homology induced by Gk/G (F q) -> Gk/G (F q A , we fit th is map and the L ang isom orphisms into the following comm utative square Gk / Q( F q ) ----------------------

i/^ ,q

i

I Gk

0 q'/q

c Gk

12. APPLICATIONS TO FIN IT E C H E V A L L EY GROUPS

where

117

0 q 7 q is the product of the maps cffl : Gk -> Gk for 0 < i < t with

q ' = q^. We readily verify that the re strictio n of 6 ^ ^

to T k induces

d * 7* : H j (T k,Z /£ ) -» H jC T ^ Z /£ ) given a s m ultiplication by 1 +q + ••• +q* 1 B ecau se

H *(T k,Z/f?) is generated by H 1( T ^ ,Z /£ ) ,

# ? '/ q - - K ( T k ’ ZM is the

we conclude that

- H / r k,z /o

O-map w henever £ divides ( q - l ) / ( q —1 ) = l + q + - " + q t _ 1 .

If H ^ T jZ /Q -> H ^ G ^ /E )

is s u rje ctiv e , the naturality of Prop osition

8 .8 im plies that ^ 7 q :H *(G k , Z / £ ) . H * ( G k,Z/l>) is a ls o the O-map for t su fficien tly large, so that colim H*(Gk/G k(F q) ,Z /Q = 0 as required.

The remaining c a s e s to co n sid er are

£ = 2 and q odd.

G = SOn o r.S p in n ,

In this c a s e ,

r f :H *(G k .Z / 2 ) _ H *(G k,Z / 2 )

is seen to be the id entity, b e ca u se

can be viewed a s the re strictio n

of : H ^ S O j Z ^ ) -> H ^ (S O ,Z /2 ), Q V f : H * (S p in ,Z /2 ) - H *(S p in ,Z /2 ) (a s in the proof of Theorem 1 2 .7 below).

T hu s,

tf * :H * (G k,Z / 2 ) -> H*(Gk,Z / 2 ) is the O-map for t even ( 0 ^ /^ *

is m ultiplication by t on prim itives in

H *(G k,Z / 2 ) ), so that colim H *(G k/G (F q); Z /2 ) - 0 . ■

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

118

If G is a reductive group over an a lg e b ra ically clo se d field of c h a ra c te r is tic G ^ (F)

p and if F = F p , then the commutator subgroup of G (F ) =

is p erfect.

We re ca ll th at the Quillen plus co n stru ctio n [43] B G (F ) -> B G (F )+

(with re sp e ct to the commutator subgroup) induces an isomorphism in homology and is the ab elian izatio n map on fundamental groups. Using a weight argument, one can e a sily verify that no cohomology c l a s s of H * (U (F ),Z /p ) G ^,

where

is invariant under T (F ) for any reductive group

is a maximal torus of G^ and

rad ical of a minimal p arab olic.

B ecau se

is the unipotent

U(Fq) con tain s a p-Sylow sub­

group of G(Fq) for any q = p ^ , we conclude that H * (G (F ),Z /p ) = 0 (c f. [64] or [34], P rop osition 4 .1 ). T his vanishing of the

Z /p-cohom ology of G (F ) and C orollary 1 2 .4

enable us to determ ine the homotopy type of B G (F )+ (a s in [3 2 ], Theorem 2 .2 ), P R O P O S I T I O N 1 2 .5 .

L e t G d en o te eith er G L n , S L n , SOn , Spinn or

Sp2n ior som e n > 0

and let F = Fp for som e prime p .

B G (F ) -> (Z /^ oo ° S in g.(B G (C ))

T h en the maps

determ in e a map ( unique up to homotopy

on fin ite sk eleta ) B G (F )+ -» w hich can be id en tifie d with the fib re

o Sin g.(B G (C )) Q /Z ^ ( B G ( C ) )

of the map

S i n g .( B G ) - .( Z (p)) ooo Sin g .(B G ), w here Z ^

is the su b rin g of Q c o n s is tin g of rationaIs w hose denom ina­

tors are not d iv is ib le by p . P ro o f. We employ D. S ullivan’s “ arithm etic fibre sq u a re ” technique to con clu d e that the vanishing of H * (B G (F ),Z /p )

and

that the maps B G (F ) -> (Z /Q ^ o S in g.(B G (C ))

H *(B G (F ),Q ) imply

12. APPLICATIONS TO FIN IT E C H E V A L L EY GROUPS

119

determ ine a map B G (F ) -> S in g .B G (C ). The uniqueness up to homotopy of th is map when re stricte d to finite sk e le ta of B G (F ) is given by ([1 3 ], VI. 8 .1 ).

B ecau se

B G (C ) is simply co n n ected , th is map fa cto rs uniquely

through a map B G (F )+ -> S in g .(B G (C )); b e ca u s e and

B G (F ) has triv ial

Z /p

Q homology, this la tter map uniquely fa cto rs through a map B G (F )+ ^ ( Q / Z (p))(B G (C )) . B y C orollary 1 2 .4 , th is map induces an isomorphism in integral

homology; b e ca u se

771(B G (F )+) is ab elian , we con clu d e th at this map

induces an isomorphism on fundamental groups.

For

G = S L n , Spinn , or

Sp2 n > B G (C ) is 2-co n n ected so that ( Q /Z ^ ) ( B G ( C ) )

is simply co n ­

n ected ; the Whitehead theorem then im plies that B G (F )+ -> ( Q / Z ^ ) ( B G ( C ) ) is a homotopy eq u iv alen ce.

B ecau se

772(S in g.(B S O n(C ))) - ^ ( ( Z ^ is su rje ctiv e for all

p,

o Sing.(BSO n(C )))

we sim ilarly con clu d e that BSOn( F ) ->

( Q /Z ^ p (BSOn(C )) is a homotopy eq u iv alen ce. F in a lly , B G L n( F ) + -* (Q /Z ^ ^ (B G L n(C ))

is a homotopy eq u iv alen ce, b e ca u s e the induced map

on universal covering s p a c e s is

B S L n( F ) + -» (Q /Z ^ p (B S L n( C ) ) . ■

In order to s ta te the most in terestin g con seq u en ce of Theorem 1 2 .2 , we must re ca ll the following definition (D efinition 1 .2 of [34]) of the c la s s i c a l C h ev a lle y gro u p s. D E F I N I T I O N 1 2 .6 .

Let

q

b e a p rim e p o w e r .

Then

F 'P q

d en o tes the

s i n g u l a r c o m p l e x of t h e h o m o to p y t h e o r e t i c f i b r e of t h e map

d ( l , ^ ) : B G ( C ) -> B G (C ) in the following four c a s e s , given with d isc re te group G(F^) (i) rep resen ts

G(Fq ) equals

G L (Fq) and d (l, *Fq L ) : B G L (C ) -> B G L (C )

on com plex K-theory.

120

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

(ii) se n ts

G (F ) eq uals l - 1?^

SO(Fq) and d ( l , ^ Q) : BSO (C) - B SO (C ) repre­

on oriented re a l K-theory.

(iii) G(Fq) equals

Sp(Fq) and d ( l , ^ s p ) : B Sp(C ) -* B Sp (C ) rep resen ts

on sym p lectic K-theory with q odd. (iv)

G(Fq ) equals

U(Fq) and d Q ,1? ^ ) : B G L (C ) -> B G L (C ) rep resen ts

l _ ^ _cl on com plex K-theory. ■ The following theorem , the fundamental resu lt of [34], e a sily d eter­ mines the K-^heories of fin ite fie ld s (the homotopy groups of B G L (F q)+ , B S O (F )+ , B S p (F )+ , and B U (F )+ ). T h e se “ u nitary” K-groups of the T. T. 4 finite field Fq have been tabulated in Theorem 1 .7 of [34]. The original com putation of ^ (B G L ^ F q )* )

w as ach ieved by D. Quillen in [64] by

other methods. T H E O R E M 1 2 .7 .

With the notation of D efin itio n 1 2 .7 , there a re homotopy

e q u iv a le n c e s a s s o c ia t e d to the sq u a res (1 2 .2 .1 ) X :B G (F q)+ - F 4 * for ea ch of the four c a s e s P roof.

G = GL , SO , S p , an d U

of D efin itio n 1 2 .1 .

One verifies th at H *(BG (Fq),Z /p ) = 0 where q is a power of the

prime p by using a vanishing range for H *(B G (Fqd ),Z /p ) as

which in c re a se s

d in cre a se s and by employing the transfer H *(BG (Fq) ,Z /p )

([3 4 ], Theorem 1 .4 ).

H *(B G (Fqd ),Z /p )

B e c a u s e both BG (Fq )+ and F 'P q are sim ple s p a c e s

(in fa c t, infinite loop s p a c e s ) with triv ial cohomology for c o e ffic ie n ts , it su ffice s to exhibit y

Z /p

and

Q

such that

X * :H * (B G (F q)+,Z /£ ) is an isomorphism for all primes £ ^ p .

H *(F «I* .Z /f ) As in the proof of P rop osition 1 2 .5 ,

we conclude that it su ffice s to exhibit maps

12. APPLICATIONS TO FIN ITE C H E V A L L EY GROUPS

121

: B G (Fq) - ( Z / O ^ F ^ for each prime £ ■/= p inducing isomorphisms X ^ H * ( ( Z / £ ) oooF ^

)Z /£ ) - H *(B G (Fq) ,Z /£ ) .

The colim it of ( 1 2 .2 .1 ) with re sp e ct to

n for

= Gn ^ equal to

G L n k , S 0 n k , Sp2n k , or G L n k and q determ ines a com m utative square BG (Fq)

( Z / £ ) „ o S ing.(B G (C ))

(1 2 .7 .1 )

(Z /£ l

( Z / £ ) „ ° S in g.(B G (C ))

o S in g.(B G (C )x 2 ) .

As is shown in P rop osition 2 .1 1 of [34], the lower horizontal arrow of (1 2 .7 .1 )

is

1 x

, sin c e

: BG n ^ -> BG n ^ and the homotopy equiva­

le n ce s of P rop osition 8 .8 determ ine maps ( Z / t ) " o S in g.(B G n(C )) - ( Z / f ) ^ o Sing.(B G n(C )) which sta b iliz e to ^

: B G (C ) -> BG (C ) (so that

( )t c ( )_1 o (^q : BGLn^k -> BGLn>k determ ines

: B G L (C ) -> B G L (C )) .

By definition of F 'P q ,

^ ts *n a homotopy c a rte s ia n square

F l^

Sing.(BG (C)>

( 1 2 . 7 .2 ) l x 'P j S in g.(B G (C ))

Sing.(BG (C)> x2

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

122

are simply co n n ected ,

(Z /l)^

B ecau se

B G (C ) and B G (C )x2

applied to

(1 2 .7 .2 )

yield s another homotopy c a rte s ia n square ( Z / l ) ^ ° (1 2 .7 .2 ) .

T herefore, we may ch o o se a map betw een sq uares (in the homotopy categ o ry ) 0 ( 1 2 .7 .1 ) - ( Z / E ) ^ o ( 1 2 .7 .2 ) with the map on upper left corner defined to be

and the map on the

other corn ers taken to be the identity. T he fa ct that X p H * ((Z /£ )oooF ' P ^ Z / 0 - H *(BG (Fq),Z /Q

is an isomorphism follows from the observation that £

induces an isomor­

phism of Eilenberg-M oore sp e ctra l seq u e n ce s (or th at £ re stricted to left v e rtica l arrows induces an isomorphism of Serre sp e ctra l se q u e n ce s). ■ An ap plication of Theorem 1 2 .7 proved by the author and S. Priddy is given in [41].

T he reader is referred to that paper for d e ta ils .

13.

FUNCTION C O M P LE X ES

R ecen t work by the author [39] and by the author and W. Dwyer [27] relatin g algeb raic K-theory to to p ological K-theory has required the use of function com p lexes of e ta le to p o lo g ical ty p e s.

T his chapter is intended

to provide the foundational m aterial n e ce s sa ry for th ose ap p licatio n s as well as th ose envisioned in the future. Much of th is ch apter (P ro p o sitio n s 1 3 .4 , 1 3 .6 , and C orollary 1 3 .7 ) is devoted to proving that maps of domain or range satisfy in g ce rtain proper­ tie s induce homotopy eq u iv alen ces of function com p lexes.

T h e se resu lts

enable one to use the fin iten ess theorem s of C hapter 7 and the com parison theorems of Chapter 8 to partially identify various function com p lexes. The s p e cific situ atio n relevant to alg eb raic K-theory is treated in P ro p o si­ tion 1 3 .1 0 , based on fin ite n e ss properties verified in C orollary 1 3 .9 .

The

relation sh ip of th e se function com p lexes to alg eb raic stru ctu res is d escrib ed in P rop osition 1 3 .2 . We re c a ll th at the function co m plex Hom(S. ,T .) sim p licial s e ts

S. and T.

(with b ase points

s

a s s o c ia te d to pointed

and t re sp e c tiv e ly ) is

the pointed sim p licial s e t w hose s e t of n-sim plices is the s e t of maps of pairs of sim p licial s e ts Homn(S. ,T ) = Hom((S. x A [ n ] ,{s } x A [n]), (T . ,t)) . F o r all pointed sim p licial s e ts

R . , S. , T . , there is a can o n ical

isomorphism Horn. (R . ,Hom. (S. ,T .) ) = Horn. (R. where R.

a

S. = R. x S. / ( i r ! x S. U R . x Is I).

then Horn. (S. ,T .)

a

If T .

S. ,T .) is a Kan com plex,

is a lso a Kan com plex so that the n-th homotopy

123

124

E T A L E HOMOTOPY OF SIMPLICIAL SCHEMES

group of the component of Horn. (S. ,T .)

containing

of homotopy c la s s e s of maps F :S . x A[n] -> T .

f : S. -> T.

c o n s is ts

re la tiv e to

f ° p r 1 U t : S. x s k j ^ A[n] U { s ! x A[n] -» T . . We extend

Horn. (

, ) to a functor

Horn. ( , ) : p ro-(s. s e t s * ) x pro-(s. s e t s * ) -> p ro-(s. s e t s * ) by defining (1 3 .1 )

Horn. (iS? ; i e l!, ! t ! ; j eJ j) = icolim {Horn. (S! , T J! ); i elS; j eJ 5 ■ I

In the following proposition, we introduce the alg eb raic function com ­ plex and re la te it to the function com plex of e ta le to p o lo g ical ty p es. P R O P O S I T I O N 1 3 .2 .

For p o in ted sim p licia l s c h e m e s

X. ,x

and Y . , y ,

we d efin e the a lg e b ra ic fu n ctio n com plex Horn. (X . ,Y .) e ( s . s e t s * ) ( with b a s e points im plicit) to b e the sim p licia l s e t w hose n -sim p lices a re maps of pairs (X . ® A [n], ix \® A[n]) -> (Y . ,y ) .

If X .

and Y .

a re locally

no etherian, then th ere is a natural map of pointed sim p licia l s e ts Horn. (X . ,Y .) -

lim

Horn. ( ( X .) e t , ( Y .)e t ) •

H R R (Y .)

P ro o f. T his map is obtained by observing that an n-sim plex of Horn. (X . ,Y .)

represented by X . ® A[n] -> Y .

determ ines

(X . ® A [n])et ->

( Y .) et in lim Hom0((X . ® A [n])e t ,( Y .)e t ) and thus in

lim Hom0(( X .) e t x A [n ]x A [n ]/S x ix A [n ],(Y .)e t )

«

lim Homn( ( X .) e t ,(Y .)e t)

by Prop osition 4 .7 . ■ As an immediate corollary of P rop osition 1 3 .2 , we con clu d e the e x is te n c e of a natural map

13. FUNCTION COM PLEXES

(1 3 .3 )

Horn. (X . ,Y .) -

holim

125

Horn. (( X .)e t ,( Y .) e t)

H R R (Y .)

thanks to the natural transform ation B ecau se

lim( ) -> holim( ) of [12], X I. 3 .5 . W. of pointed Kan com p lexes induces a pointed homotopy betw een the maps f* ,g * : Horn, ( { s !i ,T .) -> Horn. ({s !},W .) (of pointed Kan co m p lexes).

C onsequently, we im mediately conclude the

following. P r o p o s i t i o n

1 3 .4 .

L e t {SM

p ro-(s. s e t s ^ ) .

€

T h en Horn. (Is il,

)

d eterm in es a functor Horn. (IS !!,

) : pro-H^ -> pro-K^ .

In particular, if 1t!S -» {w!*} is a map in pro^Kan^) ( w here (Kan^) the fu ll su b ca teg o ry of (s . s e ts ^ ) an isom orphism in pro-K^ ,

is

co n s is tin g of Kan c o m p le x e s ) w hich is

then

Horn.

- Horn. ({SM, iW^i)

is a lso a map in pro-(KanH 0,

there is a natural isomorphism

Hom-r/ ( 2 mS. , T . ) ,

77m(Hom. (S. ,T .))

where the homotopy groups of Horn. (S. ,T .) are th ose

based a t the point map S. -> T . U A[m] x I s !) for m > 0 and pro-(s. s e ts ^ ) and T . 77m(Hom. ( l S !!,T .))

a pointed Kan com plex.

^

and where

S mS. = A[m] x S ./( s k m_ 1 A [m ]x S .

£ ° S . = S. . More gen erally, for Is ! ;i< rl! 0 .

C onseq u ently, for any {SM e p ro-(s. s e ts ^ ) and any

iT? ; je jS