Alphabeta MathNormalization Finiteness for Affine Domains
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Fields and Cyclotomic Extensions
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
Algebraic Geometry
Commutative Algebra
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Lemma
Integral closure is unchanged across an integral intermediate domain
Lemma
Finite purely inseparable rational extensions admit a finite Frobenius envelope
Lemma
Finite-variable polynomial algebras over fields are integrally closed
Lemma
Submodules of finite modules over a Noetherian ring are finite by induction
Lemma
Integral closure in a purely inseparable rational envelope is finite
Lemma
A finite normal extension is separable over its purely inseparable fixed field
Theorem
Polynomial algebras over fields have finite integral closures
Theorem
A finite-type domain over a field has finite normalization
Corollary
The normalization of an irreducible affine variety is finite
Lemma
Finite normalization commutes with principal localization
Sources
- J. S. Milne, A Primer of Commutative Algebra, v4.03, §6
- Stacks Project, Lemma 10.161.12 (Japanese rings)
- Stacks Project, Lemma 10.161.13 (polynomial N-2)
- J. S. Milne, Fields and Galois Theory, v5.10, Chapter 3
- Stacks Project, §10.37 (normal rings)
- J. S. Milne, A Primer of Commutative Algebra, §3
- Stacks Project, Lemmas 10.161.12–13 (Japanese rings)
- Stacks Project, Lemma 9.27.3 (normal extension decomposition)
- J. S. Milne, A Primer of Commutative Algebra, §6, §17
- J. S. Milne, Algebraic Geometry, Proposition 8.3 and Example 8.6
- Stacks Project, Lemma 10.36.11 (integral closure commutes with localization)