Alphabeta MathClassical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
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1 · Prerequisites
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Categories, Functors and Natural Transformations
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Linear Independence, Bases and Dimension
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinals, Cardinals, and Transfinite Recursion
- Polynomial Rings, the Division Algorithm and Roots
- Presheaves Sheaves Stalks and Sheafification
- Prime Spectra and Radicals
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Roots, Rational Powers, and Classical Inequalities
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
- Simple Field Extensions and the Construction of the Complex Numbers
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tensor Products of Modules
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Field of Fractions and Localisation
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Zariski Topology on Prime Spectra
2 · Logical flowchart
3 · Items
Definition
Classical affine algebraic sets, including the empty boundaries
Lemma
A classical zero locus depends only on the generated ideal and its radical
Theorem
Classical affine zero loci form the Zariski closed sets
Definition
The classical vanishing ideal
Lemma
Zero loci and vanishing ideals form a Galois connection
Theorem
Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals
Definition
A classical affine variety
Lemma
Irreducibility is equivalent to the nonempty-open intersection criterion
Lemma
Every nonempty open of a classical affine variety is dense
Definition
The coordinate ring of a classical affine algebraic set
Theorem
A classical affine variety has a domain coordinate ring, and conversely
Theorem
Polynomial functions on an affine algebraic set are its coordinate ring
Definition
A principal open subset of a classical affine variety
Lemma
Principal opens form a basis and multiply under intersection
Definition
A regular function on an open subset of a classical affine variety
Lemma
Classical regular functions satisfy locality and unique gluing
Theorem
Regular functions on a principal open are the principal localization
Lemma
A classical affine algebraic set has a unique finite irredundant decomposition
Definition
A reduced finitely generated k-algebra
Lemma
Classical affine points are maximal ideals
Theorem
Global regular functions on a classical affine variety are its coordinate ring
Definition
A morphism from an open subset of a classical affine variety to an affine variety
Definition
Images and set-theoretic fibres of classical regular maps
Lemma
A classical morphism pulls Zariski closed sets back to closed sets
Theorem
Classical affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms
Theorem
Classical affine algebraic sets and reduced finitely generated k-algebras are contravariantly equivalent
Lemma
Affine-source morphisms agreeing on a dense open agree everywhere
Definition
Germs and the local ring of a classical affine variety
Theorem
The classical affine local ring is localization at the point's maximal ideal
Theorem
Every nonempty principal open is a classical affine variety
Definition
A classical affine open subset and its coordinate ring
Definition
The function field of an irreducible classical affine variety
Lemma
Regular functions on a nonempty open embed in the affine function field
Theorem
The function field is independent of the chosen nonempty principal affine open
Definition
A rational map as an equivalence class of morphisms on nonempty opens
Lemma
The rational-map relation is transitive
Definition
The candidate domain of a rational map
Definition
Dominant classical morphisms and rational maps
Lemma
Morphisms defined on an open source and agreeing on a dense open agree on their common domain
Lemma
Compatible classical morphisms to an affine target glue over an open cover
Theorem
A rational map to an affine target has a unique maximal open domain
Lemma
Dominant rational maps compose on nonempty open domains
Lemma
Dominant maps pull back function fields functorially
Theorem
Dominant rational maps to an affine variety correspond to field embeddings
Definition
Integral classical varieties in the compatible affine-atlas register
Definition
Classical algebraic prevarieties, regular maps, and varieties
Lemma
Compatible affine charts of an integral classical variety have one function field
Definition
Birational maps and birational equivalence of classical varieties
Theorem
Classical integral varieties are birational exactly when their function fields are isomorphic over k
Example
The affine-line coordinate, local, and function-field dictionary
Sources
- J. S. Milne, Algebraic Geometry v6.10, §2a, pp. 36–37
- J. S. Milne, Algebraic Geometry v6.10, §2a p. 36; Theorem 2.16 preamble p. 42
- J. S. Milne, Algebraic Geometry v6.10, Proposition 2.10, pp. 38–39
- J. S. Milne, Algebraic Geometry v6.10, §2e pp. 40–42
- J. S. Milne, Algebraic Geometry v6.10, Proposition 2.10 pp. 38–39, Proposition 2.14 p. 41, and Remark 2.23 p. 44
- J. S. Milne, Algebraic Geometry v6.10, 2.13–2.17, 2.20, 2.27–2.28 and §2i, pp. 41–49
- J. S. Milne, Algebraic Geometry v6.10, §2h p. 45
- J. S. Milne, Algebraic Geometry v6.10, §2i p. 48
- J. S. Milne, Algebraic Geometry v6.10, Proposition 2.27 and §2i, pp. 45–48
- J. S. Milne, Algebraic Geometry v6.10, §2i p. 49
- J. S. Milne, Algebraic Geometry v6.10, Proposition 2.37, p. 49
- J. S. Milne, Algebraic Geometry v6.10, Definition 3.8, p. 61
- J. S. Milne, Algebraic Geometry v6.10, Proposition 3.9, p. 61
- J. S. Milne, Algebraic Geometry v6.10, Lemma 3.10 and Proposition 3.11, pp. 61–62
- J. S. Milne, Algebraic Geometry v6.10, Propositions 2.27, 2.31 and Corollary 2.32, pp. 45–47
- J. S. Milne, Algebraic Geometry v6.10, §3e pp. 65–66
- J. S. Milne, Algebraic Geometry v6.10, Example 2.13, 2.20, and §3e, pp. 41, 43, 65
- J. S. Milne, Algebraic Geometry v6.10, Proposition 3.11 final paragraph, p. 62
- J. S. Milne, Algebraic Geometry v6.10, §3d and Proposition 3.26, pp. 64–67
- J. S. Milne, Algebraic Geometry v6.10, §2j pp. 49–50
- J. S. Milne, Algebraic Geometry v6.10, Propositions 3.24–3.26, pp. 66–67
- J. S. Milne, Algebraic Geometry v6.10, §3e and Proposition 3.25, pp. 65–67
- J. S. Milne, Algebraic Geometry v6.10, Proposition 3.26 and the separatedness calculation of §5c, pp. 67, 102
- J. S. Milne, Algebraic Geometry v6.10, §3b, p. 60
- J. S. Milne, Algebraic Geometry v6.10, Corollary 3.12 and 3.17, pp. 62–64
- J. S. Milne, Algebraic Geometry v6.10, §3h pp. 71–72 and Proposition 3.11 pp. 61–62
- J. S. Milne, Algebraic Geometry v6.10, §3h pp. 71–72
- J. S. Milne, Algebraic Geometry v6.10, §2i p. 49 and §3k p. 74
- J. S. Milne, Algebraic Geometry v6.10, §3k p. 74; Definition 3.8 p. 61
- J. S. Milne, Algebraic Geometry v6.10, §3k p. 74 and Proposition 3.32 p. 71
- J. S. Milne, Algebraic Geometry v6.10, §5l p. 117
- J. S. Milne, Algebraic Geometry v6.10, Proposition 3.34 p. 72 and §§5k–l pp. 116–117
- J. S. Milne, Algebraic Geometry v6.10, Lemma 5.6 and Proposition 5.8, pp. 102–103
- J. S. Milne, Algebraic Geometry v6.10, Proposition 3.9 p. 61 and §5d pp. 103–104
- J. S. Milne, Algebraic Geometry v6.10, §5k–l pp. 116–117
- J. S. Milne, Algebraic Geometry v6.10, §5k and Proposition 5.38, pp. 116–117
- J. S. Milne, Algebraic Geometry v6.10, Proposition 5.38 p. 117; Proposition 3.34(a) p. 72
- J. S. Milne, Algebraic Geometry v6.10, Definitions 5.2, 5.7 and Proposition 5.4, pp. 100–102
- J. S. Milne, Algebraic Geometry, Definitions 5.2 and 5.7
- The Stacks Project, Definition 26.2.1
- J. S. Milne, Algebraic Geometry v6.10, 3k p. 74, 5.10 p. 103, §5k p. 116
- J. S. Milne, Algebraic Geometry v6.10, Proposition 3.36 p. 74 and Proposition 5.39 p. 117
- J. S. Milne, Algebraic Geometry v6.10, 3.17 pp. 63–64, Proposition 3.32 p. 71 and Proposition 3.26 p. 67