Alphabeta MathAlgebraic Zariski Main for Quasi-Finite Morphisms: Examples
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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
Example
The punctured affine line as an open finite factorization
Example
A finite algebra is its own Zariski Main factor
Counterexample
Quasi-finite does not imply finite
Sources
- The Stacks Project, Commutative Algebra, Section 10.123, Lemma 10.123.14 and its illustration by an open immersion
- J. S. Milne, A Primer of Commutative Algebra, version 4.03, Corollary 17.12
- The Stacks Project, Commutative Algebra, Section 10.123, Theorem 10.123.12 and Lemma 10.123.14 in the case of a finite algebra
- The Stacks Project, Commutative Algebra, Section 10.123, Theorem 10.123.12 and Lemma 10.123.14
- J. S. Milne, A Primer of Commutative Algebra, version 4.03, Aside 17.9 and Corollary 17.12