Alphabeta MathFinite Proper and Projective Morphisms — Examples
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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
- 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
- Determinants of Matrices over a Commutative Ring
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Suprema and Infima
- Topology of ℝ
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- 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
Finite power map of the affine line
Example
Closed immersion from a quotient ring
Example
Valuative extension of projective coordinates
Counterexample
The affine line is not proper
Counterexample
A nonclosed open immersion is not proper
Example
Incidence projection has closed determinantal image
Counterexample
Under AC, proper integral finite-type schemes over fields with multiple points are not affine
Counterexample
A proper nonprojective scheme from glued projective spaces
Example
The empty morphism is finite, proper and projective
Sources
- Stacks Project, Morphisms of Schemes, Definition 29.45.1 (tag 01WH)
- Ravi Vakil, Foundations of Algebraic Geometry, 2011 public draft, §8.3.6 Example 1: Branched covers
- Stacks Project, Morphisms of Schemes §§29.11, 29.42–45
- The Stacks Project, Morphisms of Schemes, Lemma 29.44.5 (tag 01WC): projective space is proper over the base, proved via the valuative criterion
- Vakil, The Rising Sea, Section 11.3.8 (the standard charts of projective space), printed pp. 309-310
- The Stacks Project, Morphisms of Schemes, Section 29.41 and Example 29.42.7
- Vakil, The Rising Sea, §11.3.1 and Exercise 11.3.A
- Stacks Project, Morphisms of Schemes, Definition 29.42.1 (tag 01W0) and §29.42
- Vakil, The Rising Sea, §11.3 discussion of properness (open immersions need not be proper)
- Vakil, The Rising Sea §§8.3, 11.3, 17.4
- The Stacks Project, Varieties, Section 33.9 (a proper variety of positive dimension is not affine)
- Vakil, The Rising Sea, Section 8.3
- Vakil, The Rising Sea, Sections 17.4.8-17.4.12 (gluing two schemes along isomorphic closed subschemes; the proper nonprojective example)
- The Stacks Project, More on Morphisms, Situation 37.67.1 (tag 0ECI) and Lemma 37.67.2 (tag 0ECJ)
- The Stacks Project, Morphisms of Schemes, Lemma 29.45.12 and Example 29.42.7
- The Stacks Project, Constructions of Schemes, Definition 27.13.1 and Lemma 27.13.3