Alphabeta MathDiagonals Separated Morphisms and Valuative Uniqueness
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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
- 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
- Tensor Products of Modules
- The Field of Fractions and Localisation
Category Theory
Commutative Algebra
Linear Algebra
Real Analysis
Scheme Theory
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
2 · Logical flowchart
3 · Items
Definition
Immersion of schemes
Definition
Separated morphism of schemes
Lemma
Closed immersions are local on the target
Definition
Separated S-scheme
Lemma
Every scheme diagonal is an immersion
Lemma
Affine morphisms are separated
Corollary
Affine schemes and affine morphisms are separated
Lemma
Separatedness survives base change
Lemma
Separated morphisms compose
Lemma
Separatedness is local on the base
Lemma
Monomorphisms and diagonals
Lemma
Closed graphs over separated targets
Theorem
Equalizers into separated schemes are closed
Corollary
Agreement on a schematically dense open
Lemma
Quasi-separatedness and the diagonal
Definition
Valuative uniqueness diagram
Lemma
Separatedness implies valuative uniqueness
Lemma
A quasi-compact immersion with nonclosed image has a boundary specialization
Lemma
A local domain has a dominating valuation overring
Lemma
An immersion with closed image is a closed immersion
Theorem
Valuative uniqueness detects separatedness
Theorem
Immersions are monomorphisms locally of finite type
Lemma
Open and closed immersions are separated
Theorem
Affine-overlap criterion for separatedness
Corollary
The affine line with doubled origin is not separated
Definition
Relative projective space from standard charts
Lemma
The relative projective-space diagonal is closed
Remark
Separated is not Zariski Hausdorff
Remark
The valuative criterion quantifies over all valuation rings
Sources
- The Stacks Project, Schemes, Definition 26.10.2(5) and Section 26.21.2, printed pp.18, 39-40
- The Stacks Project, Schemes, Definition 26.21.3 (tag 01KK), printed p.40
- The Stacks Project, Schemes, Lemma 26.4.2 (tag 01HL), printed p.5
- The Stacks Project, Schemes, Definition 26.21.3 and Lemmas 26.21.13-15, printed pp.40-42
- The Stacks Project, Schemes, Lemmas 26.21.1-2 and Section 26.10, printed pp.35, 18
- Vakil, The Rising Sea, Sections 11.3.1-2, printed pp.306-307
- The Stacks Project, Schemes, Lemmas 26.21.1 and 26.21.15, printed pp.35, 42
- Vakil, The Rising Sea, Section 11.3.4, printed p.308
- The Stacks Project, Schemes, Lemma 26.21.15, printed p.42
- The Stacks Project, Schemes, Lemma 26.21.12, printed p.41
- Vakil, The Rising Sea, Section 11.3.3, printed p.308
- The Stacks Project, Schemes, Lemmas 26.21.9 and 26.21.12, printed p.41
- The Stacks Project, Schemes, Lemmas 26.23.1-3 (tags 01L1-01L4), printed p.47
- Vakil, The Rising Sea, Section 11.2.3, printed p.306
- The Stacks Project, Schemes, Lemma 26.21.10, printed p.41
- Vakil, The Rising Sea, Section 11.3.6, printed p.309
- The Stacks Project, Schemes, Lemma 26.21.5, printed p.40
- Vakil, The Rising Sea, Section 11.4.A, printed pp.314-315
- Vakil, The Rising Sea, Section 11.4.2, printed p.315
- The Stacks Project, Schemes, Lemma 26.21.6, printed p.40
- Vakil, The Rising Sea, Section 11.2.4, printed p.306
- The Stacks Project, Schemes, Definition 26.20.3 and Section 26.22, printed pp.37, 44
- Vakil, The Rising Sea, Section 13.7.4, printed p.383
- The Stacks Project, Schemes, Lemma 26.22.1 (tag 01KZ), printed p.44
- The Stacks Project, Schemes, Lemma 26.19.7 (tag 05JL, printed p.36) and Commutative Algebra, Lemma 10.41.5 (tag 00HY, printed p.96)
- The Stacks Project, Commutative Algebra, printed p.96
- The Stacks Project, Commutative Algebra, Definition 10.50.1 and Lemmas 10.50.2-10.50.5 (tags 00I9, 00IA, 00IB, 00IC, 052K), printed p.117
- The Stacks Project, Schemes, Lemma 26.10.4, printed p.18
- The Stacks Project, Schemes, Lemma 26.22.2 (tag 01L0), printed p.44
- The Stacks Project, Example 29.52.2 (Tag 02NV), finite type need not be quasi-separated
- The Stacks Project, Morphisms of Schemes, Section 29.15 and Schemes, Section 26.23.8, printed pp.61, 48
- The Stacks Project, Schemes, Lemma 26.23.8, printed p.48
- Vakil, The Rising Sea, Section 11.3.C, printed p.308
- The Stacks Project, Schemes, Lemmas 26.21.7-8 (tags 01KP-01KQ), printed p.41
- Vakil, The Rising Sea, Sections 11.3.11-12, printed pp.311-312
- The Stacks Project, Schemes, Lemmas 26.21.7-8 and Example 26.22.2, printed pp.41-45
- Vakil, The Rising Sea, Sections 11.3.I and 13.7.C, printed pp.309, 382
- Vakil, The Rising Sea, Section 11.3.8, printed pp.309-310
- The Stacks Project, Schemes, Section 26.14.4, printed p.25
- The Stacks Project, Schemes, Lemma 26.21.8, printed p.41
- Ravi Vakil, The Rising Sea, Exercise 11.3.B and Section 10.1.2, printed p.308
- The Stacks Project, Schemes, Section 26.21 introduction, printed pp.39-40
- Ravi Vakil, The Rising Sea, Theorems 13.7.1 and 13.7.4, printed pp.381-383
- The Stacks Project, Schemes, Lemmas 26.22.1-2, printed p.44