5.3 Relation to sheaf cohomology

Čech cohomology and sheaf cohomology are two ways to study sheaves on topological spaces. They use different approaches but are closely related. For paracompact Hausdorff spaces, they give the same results.

These cohomology theories help us understand global properties of sheaves by looking at local information. They're useful in algebraic geometry and complex analysis for classifying vector bundles and studying complex manifolds.

Čech cohomology

• Čech cohomology is a cohomology theory for sheaves on a topological space, based on open covers of the space
• It provides a way to compute the cohomology groups of a sheaf using a particular open cover and the corresponding Čech complex
• Čech cohomology relates to sheaf cohomology and can be used to compute sheaf cohomology in certain cases

Relation to sheaf cohomology

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• Čech cohomology is closely related to sheaf cohomology, as it can be used to compute sheaf cohomology under certain conditions
• For a paracompact Hausdorff space and a sheaf $\mathcal{F}$, the Čech cohomology groups $\check{H}^*(X, \mathcal{F})$ are isomorphic to the sheaf cohomology groups $H^*(X, \mathcal{F})$
• The Čech-to-derived functor spectral sequence relates Čech cohomology to sheaf cohomology, providing a way to compare the two theories

Refinement of an open cover

• Refinement of an open cover is a key concept in Čech cohomology, as the cohomology groups are defined using a particular open cover
• An open cover $\mathcal{V}$ is a refinement of an open cover $\mathcal{U}$ if for every $V \in \mathcal{V}$, there exists a $U \in \mathcal{U}$ such that $V \subseteq U$
• Refinements of open covers lead to morphisms between the corresponding Čech complexes, which induce homomorphisms between the Čech cohomology groups

Cohomology of a sheaf

• Čech cohomology can be used to compute the cohomology groups of a sheaf on a topological space
• Given a sheaf $\mathcal{F}$ and an open cover $\mathcal{U}$ of a space $X$, the Čech complex $\check{C}^*(\mathcal{U}, \mathcal{F})$ is defined using the sections of $\mathcal{F}$ over intersections of open sets in $\mathcal{U}$
• The cohomology groups of the sheaf $\mathcal{F}$ with respect to the open cover $\mathcal{U}$ are the cohomology groups of the Čech complex $\check{C}^*(\mathcal{U}, \mathcal{F})$

Injective resolutions

• Injective resolutions play a role in computing Čech cohomology and relating it to sheaf cohomology
• An injective resolution of a sheaf $\mathcal{F}$ is an exact sequence $0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots$ where each $\mathcal{I}^i$ is an injective sheaf
• The Čech cohomology groups of $\mathcal{F}$ can be computed using an injective resolution, and this computation is independent of the choice of the resolution (for paracompact Hausdorff spaces)

Sheaf cohomology

• Sheaf cohomology is a cohomology theory for sheaves on a topological space, defined using derived functors
• It provides a way to study the global properties of a sheaf by measuring the obstruction to solving certain local problems globally
• Sheaf cohomology is a powerful tool in algebraic geometry and complex analysis, with applications to the classification of vector bundles and the study of invariants of complex manifolds

Definition and construction

• Sheaf cohomology groups $H^i(X, \mathcal{F})$ are defined as the right derived functors of the global sections functor $\Gamma(X, -)$ applied to a sheaf $\mathcal{F}$ on a topological space $X$
• To construct sheaf cohomology, one uses an injective resolution of the sheaf $\mathcal{F}$ and applies the global sections functor to the resolution
• The sheaf cohomology groups are the cohomology groups of the resulting complex of abelian groups

Derived functors

• Derived functors are a key concept in the construction of sheaf cohomology
• Given a left exact functor $F$ (such as the global sections functor), the right derived functors $R^iF$ measure the failure of $F$ to be exact
• Sheaf cohomology groups are defined as the right derived functors of the global sections functor, $H^i(X, \mathcal{F}) = R^i\Gamma(X, \mathcal{F})$

Long exact sequence

• Sheaf cohomology satisfies a long exact sequence, which relates the cohomology groups of a short exact sequence of sheaves

• Given a short exact sequence of sheaves $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$, there is an induced long exact sequence in cohomology: $0 \to H^0(X, \mathcal{F}) \to H^0(X, \mathcal{G}) \to H^0(X, \mathcal{H}) \to H^1(X, \mathcal{F}) \to \cdots$

• The long exact sequence is a powerful tool for computing sheaf cohomology groups and studying the relationships between sheaves

Flasque resolutions

• Flasque resolutions are a special type of resolution used in the computation of sheaf cohomology
• A sheaf $\mathcal{F}$ is flasque (or flabby) if for every open set $U \subseteq X$, the restriction map $\mathcal{F}(X) \to \mathcal{F}(U)$ is surjective
• Flasque resolutions can be used to compute sheaf cohomology because the higher cohomology groups of a flasque sheaf vanish (acyclic resolution)

Comparison of cohomology theories

• Čech cohomology and sheaf cohomology are two different cohomology theories for sheaves on a topological space
• While they are defined differently, they are closely related and, in some cases, provide the same information about a sheaf

Čech vs sheaf cohomology

• Čech cohomology is defined using open covers and the corresponding Čech complex, while sheaf cohomology is defined using derived functors
• Čech cohomology depends on the choice of an open cover, while sheaf cohomology is independent of such choices
• Sheaf cohomology is more general and can be defined for a wider class of spaces, while Čech cohomology requires the space to be paracompact Hausdorff

Isomorphism for paracompact Hausdorff spaces

• For a paracompact Hausdorff space $X$ and a sheaf $\mathcal{F}$, the Čech cohomology groups $\check{H}^*(X, \mathcal{F})$ are isomorphic to the sheaf cohomology groups $H^*(X, \mathcal{F})$
• This isomorphism allows one to compute sheaf cohomology using Čech cohomology, which can be easier in some cases
• The proof of this isomorphism involves the Čech-to-derived functor spectral sequence and the use of acyclic resolutions

Advantages and limitations

• Čech cohomology has the advantage of being more concrete and easier to compute in some cases, as it relies on open covers and the corresponding Čech complex
• Sheaf cohomology, on the other hand, is more abstract and general, applicable to a wider range of spaces and situations
• Sheaf cohomology provides a more powerful set of tools, such as the long exact sequence and the use of derived functors, which can be used to study the relationships between sheaves and compute cohomology groups

Applications

• Sheaf cohomology and Čech cohomology have numerous applications in algebraic geometry, complex analysis, and topology
• These cohomology theories provide a way to study the global properties of sheaves and extract important invariants of spaces and maps

Classification of vector bundles

• Sheaf cohomology can be used to classify vector bundles on a topological space
• The first Chern class of a line bundle is an element of the second cohomology group $H^2(X, \mathcal{O}^*)$, where $\mathcal{O}^*$ is the sheaf of nowhere-vanishing functions
• Higher-rank vector bundles can be studied using the cohomology of the sheaf of germs of $\text{GL}_n(\mathbb{C})$-valued functions

Computation of cohomology groups

• Čech cohomology and sheaf cohomology provide methods for computing the cohomology groups of a sheaf on a topological space
• These computations can be used to study the properties of the space and the sheaf, such as the existence of global sections or the obstruction to extending local sections
• Techniques such as the Mayer-Vietoris sequence, the Leray spectral sequence, and the use of acyclic resolutions can be employed to compute cohomology groups

Invariants of complex manifolds

• Sheaf cohomology is a powerful tool for studying the invariants of complex manifolds
• The Dolbeault cohomology groups $H^{p,q}(X, \mathcal{O})$, defined using the sheaf of holomorphic functions $\mathcal{O}$, provide important invariants of a complex manifold $X$
• The Hodge numbers $h^{p,q} = \dim_{\mathbb{C}} H^{p,q}(X, \mathcal{O})$ and the Euler characteristic $\chi(X, \mathcal{O}) = \sum_{p,q} (-1)^{p+q} h^{p,q}$ are examples of such invariants

Obstruction theory

• Sheaf cohomology plays a central role in obstruction theory, which studies the obstructions to solving certain geometric problems
• The vanishing of certain cohomology groups can be interpreted as the absence of obstructions to extending local solutions to global ones
• Examples include the obstruction to lifting a map between spaces, the obstruction to finding a global section of a sheaf, and the obstruction to deforming a complex structure