Alphabeta MathDiagonals Separated Morphisms and Valuative Uniqueness — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Free Modules, Exact Sequences, Projective and Injective Modules
- 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
Real Analysis
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Presheaves Sheaves Stalks and Sheafification
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
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
Example
The diagonal of the affine line
Example
The projective-line diagonal from the bihomogeneous equation
Counterexample
The doubled-origin diagonal is not closed
Counterexample
Two DVR lifts of one diagram over the doubled-origin line
Example
The graph of a polynomial map as a closed subscheme
Counterexample
A separated scheme whose point space is not Hausdorff
Example
An open immersion has valuative uniqueness but not existence
Counterexample
DVR uniqueness need not detect nonseparatedness
Sources
- The Stacks Project, Schemes, Lemma 26.21.1 (tag 01KI) and Definition 26.21.3, printed pp.39-40
- Ravi Vakil, The Rising Sea, Proposition 11.3.1, printed pp.306-307
- The Stacks Project, Schemes, Example 26.21.8 (tag 01KQ), printed p.41
- Ravi Vakil, The Rising Sea, Proposition 11.3.8, printed pp.309-310
- The Stacks Project, Schemes, Lemma 26.21.7 and Example 26.21.8, printed p.41
- Ravi Vakil, The Rising Sea, Exercise 11.3.I, printed p.309
- Ravi Vakil, The Rising Sea, Exercise 13.7.C, printed p.382
- The Stacks Project, Schemes, Lemma 26.22.2 and Example 26.22.2, printed p.44
- The Stacks Project, Schemes, Lemma 26.21.10, printed p.42
- Ravi Vakil, The Rising Sea, Proposition 11.3.6, printed p.309
- Ravi Vakil, The Rising Sea, Exercise 11.3.B, printed p.308
- The Stacks Project, Schemes, Section 26.21 introduction, printed pp.39-40
- Ravi Vakil, The Rising Sea, Theorem 13.7.4 and Exercise 13.7.A, printed p.383
- The Stacks Project, Schemes, Lemma 26.22.1 and Section 26.23, printed pp.44-45
- Ravi Vakil, The Rising Sea, Theorems 13.7.1 and 13.7.4, printed pp.381-383
- The Stacks Project, Schemes, Lemma 26.21.7 and Lemma 26.22.2, printed pp.41, 44
- The Stacks Project, Schemes, Lemma 26.21.6 (Tag 01KH)