Alphabeta MathAlgebraic Zariski Main for Quasi-Finite Morphisms
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1 · Prerequisites
Abstract Algebra
- 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
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- 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
Algebraic Geometry
Category Theory
Commutative Algebra
Linear Algebra
Real Analysis
Scheme Theory
Set Theory
2 · Logical flowchart
3 · Items
Definition
Integral elements subalgebra of an arbitrary ring map
Definition
Quasi-finiteness at a prime of a finite-type algebra
Lemma
The leading coefficient times a root is integral
Lemma
One-variable integral correction after leading-coefficient localization
Definition
Strong transcendence over a subring
Lemma
Strong transcendence descends to reduced minimal-prime quotients
Lemma
Polynomial rings over normal domains are normal
Lemma
Quasi-finite local fibres transfer through quotients and intermediate rings
Lemma
Finite algebras over a strongly transcendental variable are nowhere quasi-finite
Lemma
A quasi-finite one-generator quotient is locally its integral closure
Lemma
Conductor radical detects every polynomial coefficient
Theorem
Algebraic Zariski Main localization at a quasi-finite prime
Corollary
The quasi-finite locus of a finite-type algebra is open
Theorem
A quasi-finite algebra factors openly through a finite algebra
Corollary
Quasi-finite algebras are source locally localizations of finite algebras
Sources
- The Stacks Project, Commutative Algebra, Section 10.123, Situation 10.123.4 and its uses
- J. S. Milne, A Primer of Commutative Algebra, version 4.03, Section 17
- The Stacks Project, Commutative Algebra, Section 10.122, Lemma 10.122.2 and Definition 10.122.3
- J. S. Milne, A Primer of Commutative Algebra, version 4.03, Definition 17.3
- The Stacks Project, Commutative Algebra, Section 10.123, Lemma 10.123.1
- The Stacks Project, Commutative Algebra, Section 10.123, Lemmas 10.123.2 and 10.123.3
- The Stacks Project, Commutative Algebra, Section 10.123, Definition 10.123.7
- The Stacks Project, Commutative Algebra, Section 10.123, Lemma 10.123.8
- The Stacks Project, Commutative Algebra, Section 10.37, Lemmas 10.37.4, 10.37.6, 10.37.7 and 10.37.8
- J. S. Milne, A Primer of Commutative Algebra, version 4.03, Section 6
- The Stacks Project, Commutative Algebra, Section 10.122, Lemmas 10.122.2, 10.122.6, 10.122.7, 10.122.9 and 10.122.10
- The Stacks Project, Commutative Algebra, Section 10.123, Lemmas 10.123.8, 10.123.9 and 10.123.10 with their proofs
- The Stacks Project, Commutative Algebra, Section 10.38, Proposition 10.38.7 (going down over normal domains)
- The Stacks Project, Commutative Algebra, Section 10.123, Lemma 10.123.11 with its proof
- The Stacks Project, Commutative Algebra, Section 10.123, Situation 10.123.4 and Lemmas 10.123.5 and 10.123.6 with their proofs
- The Stacks Project, Commutative Algebra, Section 10.123, Theorem 10.123.12 (Zariski's Main Theorem) with its proof
- J. S. Milne, A Primer of Commutative Algebra, version 4.03, Section 17 (Zariski's main theorem, Theorem 17.10)
- The Stacks Project, Commutative Algebra, Section 10.123, Lemma 10.123.13 with its proof
- J. S. Milne, A Primer of Commutative Algebra, version 4.03, Corollary 17.11
- The Stacks Project, Commutative Algebra, Section 10.123, Lemma 10.123.14 with its proof
- J. S. Milne, A Primer of Commutative Algebra, version 4.03, Corollary 17.12
- The Stacks Project, Commutative Algebra, Section 10.123, Lemma 10.123.14 (1) and (2)