The Red Book of Varieties and Schemes [2 ed.] 354063293X, 9783540632931
Mumford's famous "Red Book" gives a simple, readable account of the basic objects of algebraic geometry,
206 79 2MB
English Pages 312 Year 1999
Table of contents :
Front Matter....Pages N2-V
Front Matter....Pages 1-1
Some algebra....Pages 2-7
Irreducible algebraic sets....Pages 7-15
Definition of a morphism: I....Pages 15-24
Sheaves and affine varieties....Pages 24-35
Definition of prevarieties and morphism....Pages 35-45
Products and the Hausdorff Axiom....Pages 46-55
Dimension....Pages 56-67
The fibres of a morphism....Pages 67-75
Complete varieties....Pages 75-80
Complex varieties....Pages 80-89
Front Matter....Pages 91-92
Spec (R)....Pages 93-108
The category of preschemes....Pages 108-121
Varieties are preschemes....Pages 121-131
Fields of definition....Pages 131-142
Closed subpreschemes....Pages 143-155
The functor of points of a prescheme....Pages 155-167
Proper morphisms and finite morphisms....Pages 168-176
Specialization....Pages 177-189
Front Matter....Pages 191-191
Quasi-coherent modules....Pages 193-205
Coherent modules....Pages 205-215
Front Matter....Pages 191-191
Tangent cones....Pages 215-228
Non-singularity and differentials....Pages 228-242
Étale morphisms....Pages 242-254
Uniformizing parameters....Pages 254-259
Non-singularity and the UFD property....Pages 259-271
Normal varieties and normalization....Pages 272-286
Zariski’s Main Theorem....Pages 286-295
Flat and smooth morphisms....Pages 295-308
Back Matter....Pages 309-315
Front Matter....Pages N2-V
Front Matter....Pages 1-1
Some algebra....Pages 2-7
Irreducible algebraic sets....Pages 7-15
Definition of a morphism: I....Pages 15-24
Sheaves and affine varieties....Pages 24-35
Definition of prevarieties and morphism....Pages 35-45
Products and the Hausdorff Axiom....Pages 46-55
Dimension....Pages 56-67
The fibres of a morphism....Pages 67-75
Complete varieties....Pages 75-80
Complex varieties....Pages 80-89
Front Matter....Pages 91-92
Spec (R)....Pages 93-108
The category of preschemes....Pages 108-121
Varieties are preschemes....Pages 121-131
Fields of definition....Pages 131-142
Closed subpreschemes....Pages 143-155
The functor of points of a prescheme....Pages 155-167
Proper morphisms and finite morphisms....Pages 168-176
Specialization....Pages 177-189
Front Matter....Pages 191-191
Quasi-coherent modules....Pages 193-205
Coherent modules....Pages 205-215
Front Matter....Pages 191-191
Tangent cones....Pages 215-228
Non-singularity and differentials....Pages 228-242
Étale morphisms....Pages 242-254
Uniformizing parameters....Pages 254-259
Non-singularity and the UFD property....Pages 259-271
Normal varieties and normalization....Pages 272-286
Zariski’s Main Theorem....Pages 286-295
Flat and smooth morphisms....Pages 295-308
Back Matter....Pages 309-315
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