3.1 Sheaves and sheaf cohomology

Sheaves are like mathematical spies, gathering local intel on spaces and reporting back. They're the glue that holds together bits of information, helping us understand how local properties fit into the bigger picture.

Sheaf cohomology takes this spy network to the next level. It's a powerful tool that lets us analyze spaces by looking at how information flows and connects, revealing hidden patterns and properties we might otherwise miss.

Sheaves and their properties

Definition and construction of sheaves

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• A sheaf is a tool for systematically tracking locally defined data over a topological space
• Sheaves are defined in terms of presheaves, which are contravariant functors from the category of open sets of a topological space to another category (often the category of sets, rings, modules, or groups)
• A presheaf F on a topological space X assigns to each open set U of X an object F(U) in the target category, and to each inclusion of open sets V ⊆ U a morphism F(U) → F(V) in the target category
  • The morphism F(U) → F(U) is required to be the identity
  • If W ⊆ V ⊆ U, the composition F(U) → F(V) → F(W) must equal F(U) → F(W)

Sheaf conditions and morphisms

• A sheaf is a presheaf satisfying two additional conditions:
  • Locality: If {Ui} is an open cover of U and si ∈ F(Ui) agree on overlaps, there is a unique s ∈ F(U) restricting to each si
  • Gluing: Given an open cover {Ui} of U and elements si ∈ F(Ui) that agree on overlaps, there is some s ∈ F(U) that restricts to each si
• Morphisms of sheaves are natural transformations of underlying presheaves, giving a category of sheaves on a fixed topological space
• Operations on sheaves include direct sums, products, kernels, cokernels, tensor products, and sheaf Hom

Sheaf cohomology computations

Definition and derived functors

• Sheaf cohomology extends the notion of cohomology to sheaves, assigning a sequence of abelian groups Hi(X; F) to a sheaf F on a topological space X
• The global sections functor Γ(X, -) from sheaves of abelian groups to abelian groups is left-exact
• Sheaf cohomology groups are the right derived functors of the global sections functor

Computational techniques

• Sheaf cohomology can be computed using Čech cohomology
  • Given an open cover U = {Ui} of X, the Čech complex is defined using products of F over various intersections in U
  • Hi(X; F) is the ith cohomology group of this complex
• Sheaf cohomology can also be computed using injective or flasque resolutions
  • An injective resolution is an exact sequence 0 → F → I0 → I1 → ... where each Ii is an injective sheaf
  • A flasque resolution is similar, but with each Ii flasque (sections over open sets extend to larger open sets)
  • Applying the global sections functor to an injective or flasque resolution and taking cohomology groups yields the sheaf cohomology groups Hi(X; F)
• For coherent sheaves on projective varieties, sheaf cohomology can be computed algebraically using the Serre-Grothendieck correspondence

Applications of sheaf cohomology

Studying geometry and topology

• Sheaf cohomology is a powerful tool for studying the geometry and topology of spaces through the lens of sheaves
• The dimension of the sheaf cohomology group Hi(X; F) provides information about the (non-)vanishing of global sections of F and its twists
• Sheaf cohomology can be used to define and study invariants of varieties, such as the cohomological or geometric genus of a projective variety

Ampleness and vanishing theorems

• On a projective scheme X, Serre's criterion for ampleness states that a line bundle L is ample if and only if for every coherent sheaf F on X, there exists an integer n0 such that Hi(X, F ⊗ L^n) = 0 for all i > 0 and n ≥ n0
• The Riemann-Roch theorem for curves relates the sheaf cohomology of line bundles to their degree and the genus of the curve
  • For a line bundle L on a smooth projective curve C of genus g, χ(L) = deg(L) + 1 - g, where χ(L) is the Euler characteristic h0(C, L) - h1(C, L)
• Vanishing theorems, such as the Kodaira vanishing theorem, give conditions under which certain sheaf cohomology groups vanish, providing useful information about the geometry of varieties

Sheaves vs local properties

Stalks and local behavior

• Sheaves capture local data on a space and how it patches together globally, making them well-suited for studying local properties
• The stalks of a sheaf F at points x ∈ X, denoted Fx, encode the germs of sections near x and represent the local behavior of F near x

Locally free and quasi-coherent sheaves

• A sheaf F on a space X is locally free of rank n if for each x ∈ X, there is a neighborhood U of x such that F|U is isomorphic to the free sheaf OUn
  • Locally free sheaves correspond to vector bundles
• A sheaf F on a scheme X is quasi-coherent if for each open affine subscheme Spec(A) ⊆ X, there exists an A-module M such that F|Spec(A) is isomorphic to the sheaf M~ associated to M
  • Quasi-coherent sheaves provide a connection between the local structure of schemes and modules
• A sheaf of OX-modules F on a scheme X is coherent if it is quasi-coherent and for each open affine subscheme Spec(A) ⊆ X, the corresponding A-module M is finitely generated
  • Coherent sheaves are a well-behaved class of sheaves on schemes with finiteness properties