Alphabeta MathQuasi Coherent and Coherent Sheaves and Vector Bundles — 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 over a Principal Ideal Domain and the Canonical Forms
- 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
- 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
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- 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
Quotient module sheaf and its support
Example
Restricting an associated sheaf to a localization
Example
A coherent closed-point skyscraper
Example
Twists on the two-affine projective line
Counterexample
Global sections do not determine a sheaf on P1
Counterexample
Quasi-separatedness in pushforward cannot be omitted
Counterexample
Finite type need not be locally free
Example
Fitting ideals of a diagonal two-by-two presentation
Example
The rank-zero bundle
Counterexample
Stalk and fibre are different
Sources
- The Stacks Project, Schemes, §§26.5, 26.7, 26.24
- Ravi Vakil, The Rising Sea, 29 August 2022, Chapters 6, 14, 17
- The Stacks Project, Cohomology of Schemes §30.9
- The Stacks Project, Properties of Schemes, §§28.20, 28.26
- Gao and Zhang, Lectures on Algebraic Geometry, projective-line gluing
- The Stacks Project, Examples §110.30
- The Stacks Project, More on Algebra §15.8