Alphabeta MathQuasi Coherent and Coherent Sheaves and Vector Bundles
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 24 results · all verified · 15 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
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
- 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
- Flat Smooth and Etale 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
Definition
Module sheaf on an affine scheme
Theorem
The associated module sheaf exists
Lemma
The stalk of an associated sheaf is the localisation
Lemma
Sections of the associated sheaf on basic opens
Lemma
An associated sheaf restricts to an associated sheaf on an affine open
Definition
Quasi-coherent module on a scheme
Lemma
Descent of modules on a finite principal cover
Theorem
Affine quasi-coherent sheaves are modules
Corollary
Affine quasi-coherent sheaf determined by sections
Theorem
Checking quasi-coherence on an affine cover
Theorem
Kernels and cokernels of quasi-coherent modules
Lemma
Tensor product preserves quasi-coherence
Lemma
Scheme pullback preserves quasi-coherence
Theorem
Quasi-coherence of pushforward for qcqs morphisms
Definition
Finite type and finitely presented module sheaves
Definition
Coherent module sheaves
Theorem
Coherent sheaves on a locally Noetherian scheme
Definition
Internal Hom of module sheaves
Lemma
Internal Hom from a finitely presented sheaf is quasi-coherent
Definition
Locally free sheaves of finite rank
Lemma
Dual and base change for finite locally free sheaves
Definition
Symmetric algebra of a quasi-coherent module
Lemma
Symmetric algebras are quasi-coherent and commute with pullback
Definition
Geometric vector bundle with the sections convention
Theorem
Finite locally free sheaves and geometric vector bundles
Definition
Invertible sheaves
Lemma
Dual of a line bundle is its tensor inverse
Definition
Support of a module sheaf
Theorem
Support of a finite-type quasi-coherent sheaf is closed
Definition
Fibre of a module sheaf at a point
Lemma
Geometric Nakayama for finite-type sheaves
Theorem
Openness of the finite free locus
Lemma
Fitting ideals do not depend on a presentation
Definition
Fitting ideal sheaves
Theorem
Fitting ideals control fibre generator loci
Definition
Quasi-coherent ideal sheaves
Theorem
Quasi-coherent ideals and closed subschemes, complete route
Theorem
Quasi-coherent ideals and closed subschemes
Theorem
Scheme-theoretic image of a quasi-compact morphism
Remark
Finite type need not mean coherent
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
- The Stacks Project, Constructions of Schemes §27.6
- The Stacks Project, More on Algebra §15.8
- The Stacks Project, Schemes, Section 7
- The Stacks Project, Morphisms of Schemes, Lemma 2.3
- The Stacks Project, Morphisms of Schemes, Lemma 29.6.3