Alphabeta MathSheaf Cohomology Cech Cohomology and Comparison
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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 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
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
Algebraic Geometry
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Commutative Algebra
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Categories
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Triangulated Categories
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
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
Global sections of an abelian sheaf
Lemma
Abelian sheaves form a Grothendieck category
Theorem
Enough injective abelian sheaves
Definition
Sheaf cohomology as right derived global sections
Theorem
Degree-zero sheaf cohomology is global sections
Theorem
Long exact sequence of sheaf cohomology
Lemma
Lifting a morphism from an exact complex into an injective resolution
Lemma
Variance of sheaf cohomology
Definition
Gamma-acyclic abelian sheaf
Definition
Flasque sheaf
Lemma
Injective abelian sheaves are flasque
Lemma
Flasque kernel lifts quotient sections
Theorem
Flasque abelian sheaves are Γ-acyclic
Definition
Godement resolution of an abelian sheaf
Theorem
Godement terms are flasque and compute cohomology
Definition
Ordered Čech cochain complex of a cover
Lemma
The Čech differential squares to zero
Definition
Fixed-cover Čech cohomology
Lemma
Čech H0 equals global sections
Lemma
Ordered and alternating Čech complexes agree
Definition
Refinement map of ordered open covers
Theorem
Refinement choices induce the same Čech map
Definition
Refinement-colimit Čech cohomology
Definition
Acyclic open cover for a sheaf
Lemma
Acyclic directions of the Čech–Godement double complex
Theorem
Canonical map from fixed-cover Čech to sheaf cohomology
Theorem
Leray acyclic-cover comparison
Lemma
Čech complex for a two-open cover
Theorem
Mayer–Vietoris sequence for sheaf cohomology
Theorem
Cohomology of a finite disjoint union
Theorem
A point has no higher sheaf cohomology
Definition
Cohomological dimension relative to a sheaf class
Lemma
Cofinal Čech vanishing implies derived acyclicity
Lemma
Subspaces of a Noetherian space and its compact open subsets
Lemma
Filtered colimits of sheaves and sections over compact opens
Lemma
Filtered colimits of abelian groups are exact
Lemma
Filtered colimits and sheaf cohomology on Noetherian spaces
Lemma
The subsheaf generated by a family of sections
Lemma
Locally constant functions form a sheaf with constant stalks
Lemma
Finite filtration of a generated subsheaf of the constant integer sheaf
Lemma
Extension-by-zero generators detect sheaf-cohomology vanishing
Lemma
Pushforward along a closed immersion preserves sheaf cohomology
Lemma
The constant sheaf is the sheaf of locally constant functions
Lemma
Constant sheaves on irreducible spaces are flasque and acyclic
Lemma
Extension by zero and the closed complement: a short exact sequence
Lemma
A sheaf with no stalks off a closed subset is a pushforward
Theorem
Grothendieck vanishing on a Noetherian space
Definition
Tensor product of abelian sheaves and its total complex
Lemma
Stalks, coproducts and right exactness of the abelian sheaf tensor product
Definition
Flat abelian sheaves
Lemma
Flatness criteria and canonical epimorphisms from flat abelian sheaves
Definition
K-flat complexes of abelian sheaves in the bounded-above setting
Lemma
K-flat sheaf complexes preserve quasi-isomorphisms
Lemma
Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes
Lemma
Derived tensor product of abelian sheaves
Lemma
Morphisms from the constant sheaf are global sections
Lemma
Sheaf cohomology classes as derived morphisms
Lemma
Associator, symmetry and unitors of the abelian sheaf tensor product
Lemma
Koszul coherence of derived sheaf tensor
Definition
Cup product in sheaf cohomology
Theorem
Cup-product laws
Remark
Fixed-cover Čech can miss derived cohomology
Remark
Spectral-sequence algebra is external to this pair
Sources
- The Stacks Project, Cohomology of Sheaves
- Jiahui Gao and Shuwu Zhang, Lectures on Algebraic Geometry
- The Stacks Project, Injectives (tag 01DF)
- The Stacks Project, Derived Categories, Section 13.18: Injective resolutions (tags 013P, 013R)
- The Stacks Project, Section 6.31 (Tag 009Z), extension by zero adjunction
- The Stacks Project, Topology (Section 5.9, Noetherian topological spaces)
- The Stacks Project, Sheaves on Spaces
- The Stacks Project, Algebra (filtered colimits of modules)
- The Stacks Project, Modules on Sites (and Sheaves on Spaces, Section 29)
- The Stacks Project, Derived Categories
- The Stacks Project, Algebra
- The Stacks Project, More on Algebra