11.1 Sheaves and sheaf cohomology

Sheaves and sheaf 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

• A sheaf $F$ on a topological space $X$ consists of:
  • A set $F(U)$ for each open set $U$ in $X$
  • Restriction maps $res_{U,V}: F(U) \rightarrow F(V)$ for each inclusion $V \subseteq U$ of open sets
• These sets and maps must satisfy the following conditions:
  • (Identity) For every open set $U$, the restriction map $res_{U,U}$ is the identity map on $F(U)$
  • (Composition) If $W \subseteq V \subseteq U$ are open sets, then $res_{V,W} \circ res_{U,V} = res_{U,W}$
  • (Gluing) If $\{U_i\}$ is an open cover of an open set $U$ and $s_i \in F(U_i)$ are elements such that $res_{U_i,U_i \cap U_j}(s_i) = res_{U_j,U_i \cap U_j}(s_j)$ for all $i,j$, then there exists a unique $s \in F(U)$ such that $res_{U,U_i}(s) = s_i$ for all $i$

Morphisms, Abelian Category, and Stalks

• A morphism $\phi: F \rightarrow G$ between sheaves on $X$ is a collection of maps $\phi_U: F(U) \rightarrow G(U)$ 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 $V \subseteq U$ of open sets, the restriction map $res_{U,V}$ is surjective
  • Flasque sheaves are used in the construction of injective resolutions
• The stalk of a sheaf $F$ at a point $x \in X$, denoted $F_x$, 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 injective sheaf is a sheaf $I$ such that for any monomorphism $\phi: F \rightarrow G$ and any morphism $\psi: F \rightarrow I$, there exists a morphism $\chi: G \rightarrow I$ such that $\chi \circ \phi = \psi$
  • Injective sheaves are analogous to injective modules in homological algebra
• An injective resolution of a sheaf $F$ is an exact sequence $0 \rightarrow F \rightarrow I^0 \rightarrow I^1 \rightarrow ...$, where each $I^i$ is an injective sheaf
  • Injective resolutions are used to define sheaf cohomology groups

Definition and Computation of Sheaf Cohomology

• Given a sheaf $F$ on a topological space $X$, the sheaf cohomology groups $H^i(X, F)$ are defined as the right derived functors of the global sections functor $\Gamma(X, -)$
• To compute the sheaf cohomology groups:
  1. Choose an injective resolution $0 \rightarrow F \rightarrow I^0 \rightarrow I^1 \rightarrow ...$ of $F$
  2. Apply the global sections functor to obtain a cochain complex $0 \rightarrow \Gamma(X, I^0) \rightarrow \Gamma(X, I^1) \rightarrow ...$
  3. The cohomology groups of this cochain complex are the sheaf cohomology groups $H^i(X, F)$
• 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 $H^i(X, A)$ are isomorphic to the singular cohomology groups $H^i(X, A)$ 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 $H^0(X, F)$ is isomorphic to the set of global sections $\Gamma(X, F)$
  • Global sections are the elements that are compatible with all restriction maps
• On a contractible space $X$, the higher sheaf cohomology groups $H^i(X, F)$ vanish for any sheaf $F$ and $i > 0$
  • Contractible spaces (such as the real line $\mathbb{R}$) have trivial higher cohomology
• The first sheaf cohomology group $H^1(X, F)$ 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-derived functor spectral sequence relates Čech cohomology 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 cohomological data

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 manifold $X$, the de Rham cohomology groups (defined using differential forms) are isomorphic to the sheaf cohomology groups of the constant sheaf $\mathbb{R}$ 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 $H^i(X, E)$ 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