9.1 Sheaf cohomology

Sheaf cohomology is a powerful tool in algebraic topology that measures how local data on a space fits together globally. It extends the idea of sheaves, which track local information, to capture more complex relationships between local and global properties.

This topic introduces key concepts like derived functors and Čech cohomology, which are essential for computing sheaf cohomology. We'll explore how these methods reveal deep insights about topological spaces and their associated sheaves.

Sheaf Theory Fundamentals

Sheaf and Presheaf Concepts

• Sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space
• Consists of a topological space $X$ and a sheaf $F$ on $X$ associating data to each open set $U \subseteq X$ in a way that is compatible with restriction
• Sheaf axioms ensure local properties can be uniquely glued together
• Presheaf is a similar concept to a sheaf but without the gluing axiom
  • Associates data to open sets of a topological space and is compatible with restriction
  • May not allow for unique gluing of local sections

Étalé Space and Sheaf of Sections

• Étalé space is a topological space $E$ along with a local homeomorphism $p: E \to X$
  • Provides a useful perspective for studying sheaves ($p^{-1}(U)$ corresponds to $F(U)$)
  • Allows for visualizing how the data varies over the base space $X$
• Sheaf of sections is a fundamental example of a sheaf
  • Given a continuous map $f: Y \to X$, the sheaf of sections associates to each open set $U \subseteq X$ the set of continuous sections of $f$ over $U$
  • Encodes the idea of continuously varying data over a space

Cohomology Constructions

Derived Functor and Čech Cohomology

• Derived functor cohomology is a general method for constructing cohomology theories
  • Starts with a left exact functor $F$ (such as the global section functor $\Gamma$) and derives right derived functors $R^iF$
  • The $i$-th derived functor $R^iF$ applied to a sheaf $F$ gives the $i$-th cohomology group $H^i(X, F)$
• Čech cohomology is a concrete realization of sheaf cohomology using open covers
  • Given an open cover $\mathcal{U} = \{U_i\}$ of $X$, defines cochains as functions on intersections of open sets
  • The Čech complex gives a flasque resolution of the sheaf, allowing for computation of cohomology

Injective Resolutions and Cohomology Groups

• Injective resolution is a tool for computing derived functor cohomology
  • An injective resolution of a sheaf $F$ is an exact sequence $0 \to F \to I^0 \to I^1 \to \cdots$ where each $I^i$ is an injective sheaf
  • Existence of injective resolutions allows for the computation of $R^iF$ as the cohomology of the complex $F(I^\bullet)$
• Cohomology groups $H^i(X, F)$ are the central objects of study in sheaf cohomology
  • Measure the global obstruction to solving a geometric problem related to the sheaf $F$ (such as finding global sections)
  • Encode topological and geometric information about the space $X$ and the sheaf $F$

Special Sheaves

Flasque and Acyclic Sheaves

• Flasque sheaf is a special type of sheaf with vanishing higher cohomology
  • Defined by the property that any section over an open set can be extended to a global section
  • Flasque sheaves are used in constructing resolutions to compute cohomology
  • Examples include the sheaf of locally constant functions and the sheaf of sections of a vector bundle
• Acyclic sheaf is a sheaf that is "invisible" to the cohomology functor
  • A sheaf $F$ is acyclic if $H^i(X, F) = 0$ for all $i > 0$
  • Acyclic sheaves play a role in the machinery of spectral sequences and hypercohomology
  • Examples include flasque sheaves and soft sheaves (sheaves where sections extend over closed subsets)