Sheaves and cohomology are powerful tools in algebraic topology. They provide a way to study local-to-global properties of spaces, connecting local data to global information. This framework is crucial for understanding complex geometric and topological structures.
Sheaf cohomology generalizes other cohomology theories, offering a unified approach to various mathematical concepts. It's particularly useful in algebraic geometry, allowing us to extract meaningful information from geometric objects and their relationships.
Sheaves on Topological Spaces
Definition and Properties of Sheaves
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A sheaf F on a X consists of:
A set F(U) for each open set U in X
Restriction maps resU,V:F(U)→F(V) for each inclusion V⊆U of open sets
These sets and maps must satisfy the following conditions:
(Identity) For every open set U, the resU,U is the identity map on F(U)
(Composition) If W⊆V⊆U are open sets, then resV,W∘resU,V=resU,W
(Gluing) If {Ui} is an open cover of an open set U and si∈F(Ui) are elements such that resUi,Ui∩Uj(si)=resUj,Ui∩Uj(sj) for all i,j, then there exists a unique s∈F(U) such that resU,Ui(s)=si for all i
Morphisms, Abelian Category, and Stalks
A morphism ϕ:F→G between sheaves on X is a collection of maps ϕU:F(U)→G(U) for each open set U, compatible with the restriction maps
Sheaves form an abelian category, with kernels, cokernels, and exact sequences defined pointwise
This allows for the use of homological algebra techniques in the study of sheaves
A sheaf F is flasque (flabby) if for every inclusion V⊆U of open sets, the restriction map resU,V is surjective
Flasque sheaves are used in the construction of injective resolutions
The stalk of a sheaf F at a point x∈X, denoted Fx, is the direct limit of the sets F(U) over all open neighborhoods U of x, with the induced restriction maps
Stalks provide local information about the sheaf at each point
Sheaf Cohomology Groups
Injective Sheaves and Resolutions
An is a sheaf I such that for any monomorphism ϕ:F→G and any morphism ψ:F→I, there exists a morphism χ:G→I such that χ∘ϕ=ψ
Injective sheaves are analogous to injective modules in homological algebra
An of a sheaf F is an exact sequence 0→F→I0→I1→..., where each Ii is an injective sheaf
Injective resolutions are used to define sheaf
Definition and Computation of Sheaf Cohomology
Given a sheaf F on a topological space X, the sheaf cohomology groups Hi(X,F) are defined as the right derived functors of the functor Γ(X,−)
To compute the sheaf cohomology groups:
Choose an injective resolution 0→F→I0→I1→... of F
Apply the global sections functor to obtain a cochain complex 0→Γ(X,I0)→Γ(X,I1)→...
The cohomology groups of this cochain complex are the sheaf cohomology groups Hi(X,F)
The sheaf cohomology groups are independent of the choice of injective resolution
Computing Sheaf Cohomology
Simple Cases and Interpretations
For a constant sheaf A on a topological space X, the sheaf cohomology groups Hi(X,A) are isomorphic to the singular cohomology groups Hi(X,A) with coefficients in the abelian group A
This allows for the computation of sheaf cohomology using singular cohomology in certain cases
For a locally constant sheaf F on a locally connected space X, the sheaf cohomology group H0(X,F) is isomorphic to the set of global sections Γ(X,F)
Global sections are the elements that are compatible with all restriction maps
On a contractible space X, the higher sheaf cohomology groups Hi(X,F) vanish for any sheaf F and i>0
Contractible spaces (such as the real line R) have trivial higher cohomology
The first sheaf cohomology group H1(X,F) classifies the isomorphism classes of F-torsors on X, which are sheaves of sets locally isomorphic to F
F-torsors can be thought of as twisted versions of the sheaf F
Čech-to-Derived Functor Spectral Sequence
The Čech-to- spectral sequence relates and sheaf cohomology
It provides a way to compute sheaf cohomology using Čech cohomology of a cover and the higher direct images of the sheaf
The higher direct images measure the failure of the sheaf to be acyclic on the intersections of the cover
The spectral sequence starts with the Čech cohomology groups and converges to the sheaf cohomology groups
This allows for the computation of sheaf cohomology in terms of simpler
Sheaf Cohomology vs Other Theories
Generalizations and Connections
Sheaf cohomology generalizes singular cohomology and Čech cohomology
For a constant sheaf, sheaf cohomology recovers singular cohomology
For a good cover, sheaf cohomology is isomorphic to Čech cohomology
De Rham's theorem states that for a smooth X, the groups (defined using differential forms) are isomorphic to the sheaf cohomology groups of the constant sheaf R on X
This establishes a connection between differential geometry and sheaf theory
Applications to Vector Bundles
Sheaf cohomology can be used to compute the cohomology of vector bundles
For a vector bundle E on a topological space X, the sheaf cohomology groups Hi(X,E) of the sheaf of sections of E are isomorphic to the cohomology groups of E
This provides a sheaf-theoretic approach to studying vector bundles
The cohomology groups of vector bundles have important applications in geometry and physics
They classify topological invariants (characteristic classes) and measure obstructions to the existence of global sections