Algebraic Geometry
Mar 19 2017 algebraic geometry regular (polynomial) functions algebraic varieties topology continuous functions topological spaces.
Algebraic Geometry
Algebraic Geometry. J.S. Milne. Version 5.10. March 19 2008. A more recent version of these notes is available at www.jmilne.org/math/
THE RISING SEA Foundations of Algebraic Geometry
Foundations of Algebraic Geometry math216.wordpress.com. November 18 2017 draft c? 2010–2017 by Ravi Vakil. Note to reader: the index and formatting have
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Jan 28 2008 The aim of these notes is to develop the theory of algebraic curves from the viewpoint of modern algebraic geometry
Enumerative Algebraic Geometry of Conics
If so how many are there? Problems that ask for the number of geometric objects with given properties are known as enumera- tive problems in algebraic geometry
Algebraic Geometry
Commutative Algebra. 170 BREDON. Sheaf Theory. 2nd ed. 142 LANG. Real and Functional Analysis. 171 PETERSEN. Riemannian Geometry.
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THE HISTORICAL DEVELOPMENT OF ALGEBRAIC GEOMETRY - J
Modern algebraic geometry has deservedly been considered for a long time as an exceedingly complex part of mathematics drawing practically on every other
Enumerative Algebraic Geometry of Conics
If so how many are there? Problems that ask for the number of geometric objects with given properties are known as enumera- tive problems in algebraic geometry
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18 nov 2017 · Chapter 3 Toward affine schemes: the underlying set and topological space 99 3 1 Toward schemes 99 3 2 The underlying set of affine
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THE RISING SEA
Foundations of Algebraic Geometry
math216.wordpress.comNovember 18, 2017 draft
c ⃝2010-2017 by Ravi Vakil. Note to reader: the index and formatting have yet to be properly dealt with. There remain many issues still to be dealt with in the main part of the notes (including many of your corrections and suggestions).Contents
Preface
110.1. For the reader
120.2. For the expert
160.3. Background and conventions
170.4.⋆⋆The goals of this book
18Part I. Preliminaries
21Chapter 1. Some category theory
231.1. Motivation
231.2. Categories and functors
251.3. Universal properties determine an object up to unique isomorphism
311.4. Limits and colimits
391.5. Adjoints
431.6. An introduction to abelian categories
471.7.⋆Spectral sequences
57Chapter 2. Sheaves
712.1. Motivating example: The sheaf of differentiable functions
712.2. Definition of sheaf and presheaf
732.3. Morphisms of presheaves and sheaves
782.4. Properties determined at the level of stalks, and sheafification
822.5. Recovering sheaves from a sheaf on a base"
862.6. Sheaves of abelian groups, andOX-modules, form abelian categories
892.7. The inverse image sheaf
92Part II. Schemes
97Chapter 3. Toward affine schemes: the underlying set, and topological space 99
3.1. Toward schemes
993.2. The underlying set of affine schemes
1013.3. Visualizing schemes I: generic points
1133.4. The underlying topological space of an affine scheme
1153.5. A base of the Zariski topology on SpecA: Distinguished open sets
1183.6. Topological (and Noetherian) properties
1193.7. The functionI(), taking subsets of SpecAto ideals ofA
127Chapter 4. The structure sheaf, and the definition of schemes in general 129
4.1. The structure sheaf of an affine scheme
1294.2. Visualizing schemes II: nilpotents
1333
4.3. Definition of schemes
1364.4. Three examples
1394.5. Projective schemes, and the Proj construction
145Chapter 5. Some properties of schemes
1535.1. Topological properties
1535.2. Reducedness and integrality
1555.3. Properties of schemes that can be checked affine-locally"
1575.4. Normality and factoriality
1615.5. The crucial points of a scheme that control everything: Associated
points and primes 166Part III. Morphisms
175Chapter 6. Morphisms of schemes
1776.1. Introduction
1776.2. Morphisms of ringed spaces
1786.3. From locally ringed spaces to morphisms of schemes
1806.4. Maps of graded rings and maps of projective schemes
1866.5. Rational maps from reduced schemes
1886.6.⋆Representable functors and group schemes
1946.7.⋆⋆The Grassmannian (initial construction)
199Chapter 7. Useful classes of morphisms of schemes
2017.1. An example of a reasonable class of morphisms: Open embeddings
2017.2. Algebraic interlude: Lying Over and Nakayama
2037.3. A gazillion finiteness conditions on morphisms
2077.4. Images of morphisms: Chevalley"s Theorem and elimination theory
216Chapter 8. Closed embeddings and related notions
2258.1. Closed embeddings and closed subschemes
2258.2. More projective geometry
2308.3. The (closed sub)scheme-theoretic image
2368.4. Effective Cartier divisors, regular sequences and regular embeddings
240Chapter 9. Fibered products of schemes, and base change 247
9.1. They exist
2479.2. Computing fibered products in practice
2539.3. Interpretations: Pulling back families, and fibers of morphisms
2569.4. Properties preserved by base change
2629.5.⋆Properties not preserved by base change, and how to fix them
2639.6. Products of projective schemes: The Segre embedding
2719.7. Normalization
273Chapter 10. Separated and proper morphisms, and (finally!) varieties 279
10.1. Separated morphisms (and quasiseparatedness done properly)
27910.2. Rational maps to separated schemes
28910.3. Proper morphisms
293Part IV. Geometric" properties: Dimension and smoothness 301
Chapter 11. Dimension
30311.1. Dimension and codimension
30311.2. Dimension, transcendence degree, and Noether normalization
30711.3. Codimension one miracles: Krull"s and Hartogs"s Theorems
31511.4. Dimensions of fibers of morphisms of varieties
32211.5.⋆⋆Proof of Krull"s Principal Ideal and Height Theorems
327Chapter 12. Regularity and smoothness
33112.1. The Zariski tangent space
33112.2. Regularity, and smoothness over a field
33712.3. Examples
34212.4. Bertini"s Theorem
34612.5. Another (co)dimension one miracle: Discrete valuation rings
34912.6. Smooth (and
´etale) morphisms (first definition)
35412.7.⋆Valuative criteria for separatedness and properness
35812.8.⋆More sophisticated facts about regular local rings
36212.9.⋆Filtered rings and modules, and the Artin-Rees Lemma
364Part V. Quasicoherent sheaves
367Chapter 13. Quasicoherent and coherent sheaves
36913.1. Vector bundles and locally free sheaves
36913.2. Quasicoherent sheaves
37513.3. Characterizing quasicoherence using the distinguished affine base
37713.4. Quasicoherent sheaves form an abelian category
38113.5. Module-like constructions
38313.6. Finite type and coherent sheaves
38613.7. Pleasant properties of finite type and coherent sheaves
38913.8.⋆⋆Coherent modules over non-Noetherian rings
393Chapter 14. Line bundles: Invertible sheaves and divisors 397
14.1. Some line bundles on projective space
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