10.2 Sheaf cohomology

Sheaf cohomology is a powerful tool in mathematics that bridges local and global properties of spaces. It assigns algebraic data to open sets, allowing us to study how local information fits together to form a coherent whole.

This theory is crucial for understanding topological spaces, manifolds, and algebraic varieties. It provides a framework for measuring obstructions to extending local data globally, connecting various branches of mathematics like algebraic geometry and complex analysis.

Sheaves on topological spaces

• Sheaves are a central concept in cohomology theory that allow for the study of local-to-global properties of spaces
• They provide a way to assign algebraic data (such as functions or sections) to open sets of a topological space in a consistent manner

Presheaves vs sheaves

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• Presheaves are a more general notion than sheaves, assigning data to open sets without requiring certain compatibility conditions
• Sheaves satisfy the gluing axiom, ensuring that local sections can be uniquely patched together to form global sections
• The sheaf condition also requires that sections can be restricted to smaller open sets in a way that is compatible with the presheaf structure
• Examples of sheaves include the sheaf of continuous functions and the sheaf of smooth functions on a manifold

Sheafification of presheaves

• Sheafification is a process that turns a presheaf into a sheaf by enforcing the gluing and local identity axioms
• It is achieved by adding new sections to the presheaf that are obtained by gluing together compatible local sections
• The sheafification functor is left adjoint to the forgetful functor from sheaves to presheaves
• Sheafification allows for the extension of presheaf-theoretic constructions to the realm of sheaves

Sheaf of continuous functions

• The sheaf of continuous functions assigns to each open set $U$ in a topological space $X$ the set of continuous functions $f: U \to \mathbb{R}$
• Restriction maps are given by restricting a continuous function to a smaller open set
• This sheaf encodes the local nature of continuity, as a function is continuous if and only if it is continuous when restricted to any open cover
• The sheaf of continuous functions is a fundamental example in sheaf theory and plays a crucial role in many applications

Sheaf of differentiable functions

• On a smooth manifold $M$, the sheaf of differentiable functions assigns to each open set $U$ the set of smooth (infinitely differentiable) functions $f: U \to \mathbb{R}$
• Restriction maps are given by restricting a smooth function to a smaller open set
• This sheaf captures the local nature of differentiability and is essential in the study of differential geometry and analysis on manifolds
• The sheaf of differentiable functions is a fine sheaf, meaning it admits partitions of unity, which is a key property in many constructions and proofs

Čech cohomology

• Čech cohomology is a cohomology theory for sheaves that is based on open covers of a topological space
• It provides a way to measure the global consistency of local data encoded by a sheaf

Čech cohomology of presheaves

• Čech cohomology can be defined for presheaves by considering alternating cochains on open covers
• The cohomology groups are obtained by taking the quotient of cocycles (cochains satisfying a certain condition) by coboundaries (cochains that are the difference of two others)
• Čech cohomology of presheaves is functorial with respect to refinement of open covers
• Presheaves that are not sheaves can have non-trivial higher Čech cohomology groups

Čech cohomology of sheaves

• When applied to sheaves, Čech cohomology has better properties and is often easier to compute than for general presheaves
• The Čech cohomology groups of a sheaf $\mathcal{F}$ on a space $X$ are denoted by $\check{H}^p(X, \mathcal{F})$
• For a sheaf, the Čech cohomology groups are isomorphic to the derived functor cohomology groups (see below)
• Čech cohomology of sheaves is invariant under refinement of open covers, which is a key property for proving independence of the choice of cover

Refinement of open covers

• A refinement of an open cover $\mathcal{U} = \{U_i\}$ is another open cover $\mathcal{V} = \{V_j\}$ such that each $V_j$ is contained in some $U_i$
• Refinements allow for the comparison of Čech cochains and cohomology groups defined with respect to different covers
• A sheaf is called acyclic with respect to an open cover if its higher Čech cohomology groups vanish for that cover
• Fine sheaves, such as the sheaf of smooth functions on a manifold, are acyclic with respect to any open cover

Čech-to-derived functor spectral sequence

• The Čech-to-derived functor spectral sequence is a tool that relates Čech cohomology to the derived functor cohomology of a sheaf
• It arises from a double complex that combines Čech cochains and injective resolutions of the sheaf
• The spectral sequence converges to the derived functor cohomology groups, with the Čech cohomology groups appearing on the $E_2$ page
• In many cases, the spectral sequence degenerates at the $E_2$ page, yielding an isomorphism between Čech and derived functor cohomology

Sheaf cohomology via derived functors

• Derived functor cohomology is another approach to defining cohomology groups for sheaves, using the machinery of homological algebra
• It is based on the idea of deriving the global sections functor, which is not exact, to obtain a sequence of functors that measure the obstruction to exactness

Injective resolutions of sheaves

• 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 $\mathcal{I}^p$ is an injective sheaf
• Injective sheaves are analogous to injective modules in homological algebra and have the property that the global sections functor is exact on them
• Every sheaf admits an injective resolution, which is unique up to homotopy equivalence
• Injective resolutions allow for the construction of derived functors and the computation of sheaf cohomology

Global sections functor

• The global sections functor $\Gamma(X, -)$ takes a sheaf $\mathcal{F}$ on a topological space $X$ and returns the set (or module) of global sections $\Gamma(X, \mathcal{F})$
• Global sections are the sections of $\mathcal{F}$ defined on the entire space $X$
• The global sections functor is left exact but not right exact, meaning it preserves kernels but not cokernels
• This failure of exactness is measured by the higher derived functors of $\Gamma(X, -)$, which define sheaf cohomology

Higher direct images

• For a continuous map $f: X \to Y$ between topological spaces, the higher direct image functors $R^pf_*$ are the derived functors of the direct image functor $f_*$
• The direct image functor $f_*$ takes a sheaf $\mathcal{F}$ on $X$ and returns the sheaf $f_*\mathcal{F}$ on $Y$ whose sections on an open set $V \subset Y$ are given by $\Gamma(f^{-1}(V), \mathcal{F})$
• The higher direct images measure the obstruction to the exactness of the direct image functor
• They are related to the cohomology of the fibers of the map $f$ and play a crucial role in the Leray spectral sequence

Derived functors of global sections

• The derived functors of the global sections functor $\Gamma(X, -)$ are denoted by $H^p(X, -)$ and define the sheaf cohomology groups
• To compute $H^p(X, \mathcal{F})$, one takes an injective resolution $0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots$, applies the global sections functor to obtain a complex $0 \to \Gamma(X, \mathcal{I}^0) \to \Gamma(X, \mathcal{I}^1) \to \cdots$, and takes the cohomology of this complex at the $p$-th position
• The derived functor cohomology groups are independent of the choice of injective resolution and are functorial with respect to sheaf morphisms
• In many cases, the derived functor cohomology groups agree with the Čech cohomology groups, providing a more intrinsic definition of sheaf cohomology

Cohomology of sheaves on manifolds

• When studying sheaves on smooth manifolds, there are several important cohomology theories that relate to the underlying differential structure
• These cohomology theories often have a more geometric or analytic flavor and can be used to study properties of the manifold itself

De Rham theorem for sheaf cohomology

• The de Rham theorem states that the sheaf cohomology groups of the constant sheaf $\underline{\mathbb{R}}$ on a smooth manifold $M$ are isomorphic to the de Rham cohomology groups of $M$
• The de Rham cohomology groups are defined using differential forms and the exterior derivative, capturing the differential structure of the manifold
• This isomorphism provides a link between the algebraic notion of sheaf cohomology and the analytic notion of de Rham cohomology
• The proof of the de Rham theorem involves constructing a resolution of the constant sheaf using the sheaves of differential forms and showing that it computes both sheaf and de Rham cohomology

Poincaré lemma for sheaves

• The Poincaré lemma is a local statement about the exactness of the de Rham complex on a contractible open set in a manifold
• It states that on a contractible open set, every closed differential form is exact, meaning it is the exterior derivative of another form
• In the language of sheaves, the Poincaré lemma says that the sheaf of closed differential forms is locally exact, or a soft sheaf
• This local exactness is a key ingredient in the proof of the de Rham theorem and the comparison of sheaf and de Rham cohomology

Dolbeault cohomology of sheaves

• Dolbeault cohomology is a cohomology theory for sheaves on complex manifolds that takes into account the complex structure
• It is defined using the Dolbeault complex, which involves the $\bar{\partial}$ operator acting on $(p,q)$-forms
• The Dolbeault cohomology groups $H^{p,q}(X, \mathcal{F})$ of a sheaf $\mathcal{F}$ on a complex manifold $X$ measure the obstruction to solving the $\bar{\partial}$ equation with values in $\mathcal{F}$
• Dolbeault cohomology is related to the sheaf cohomology of the sheaf of holomorphic sections of a holomorphic vector bundle and plays a central role in complex geometry

Comparison of sheaf cohomologies

• There are various comparison theorems that relate different sheaf cohomology theories on manifolds
• The Dolbeault theorem states that the Dolbeault cohomology groups of the constant sheaf $\underline{\mathbb{C}}$ on a complex manifold are isomorphic to the sheaf cohomology groups with complex coefficients
• The Hodge theorem provides a decomposition of the sheaf cohomology groups of the constant sheaf on a compact Kähler manifold into a direct sum of Dolbeault cohomology groups
• These comparison theorems highlight the interplay between the different structures on a manifold (smooth, complex, Kähler) and the corresponding cohomology theories

Applications of sheaf cohomology

• Sheaf cohomology has numerous applications in various branches of mathematics, including algebraic and differential geometry, complex analysis, and mathematical physics
• It provides a powerful tool for studying global properties of spaces and the behavior of functions and sections on them

Serre duality for sheaves

• Serre duality is a fundamental duality theorem in sheaf theory that relates the cohomology of a coherent sheaf on a projective variety to the cohomology of its dual sheaf
• In its simplest form, for a coherent sheaf $\mathcal{F}$ on an $n$-dimensional projective variety $X$, Serre duality states that there are isomorphisms $H^i(X, \mathcal{F}) \cong H^{n-i}(X, \mathcal{F}^* \otimes \omega_X)^*$, where $\omega_X$ is the canonical sheaf of $X$
• Serre duality has numerous applications in algebraic geometry, including the study of curves, surfaces, and moduli spaces
• It is a key ingredient in the proof of the Riemann-Roch theorem for surfaces and the construction of the Picard scheme

Hodge decomposition for sheaf cohomology

• The Hodge decomposition is a decomposition of the sheaf cohomology groups of the constant sheaf on a compact Kähler manifold into a direct sum of Dolbeault cohomology groups
• It states that $H^k(X, \underline{\mathbb{C}}) \cong \bigoplus_{p+q=k} H^{p,q}(X)$, where $H^{p,q}(X)$ are the Dolbeault cohomology groups of $X$
• The Hodge decomposition is a consequence of the $\partial\bar{\partial}$-lemma and the Kähler identities, which relate the Laplacian operators associated to the Dolbeault complex and the de Rham complex
• It has important applications in the study of the topology of Kähler manifolds and the variation of Hodge structures

Characteristic classes via sheaf cohomology

• Characteristic classes are topological invariants associated to vector bundles that measure the twisting or non-triviality of the bundle
• Many characteristic classes, such as Chern classes, can be defined using sheaf cohomology
• The Chern classes of a complex vector bundle $E$ are elements of the sheaf cohomology groups $H^{2k}(X, \underline{\mathbb{Z}})$ of the base space $X$, and they measure the obstruction to the existence of global sections of $E$
• Sheaf-theoretic constructions of characteristic classes provide a unified framework for studying their properties and relationships, such as the splitting principle and the Whitney product formula

Grothendieck's algebraic de Rham theorem

• Grothendieck's algebraic de Rham theorem is a generalization of the classical de Rham theorem to the setting of algebraic geometry
• It states that for a smooth algebraic variety $X$ over a field of characteristic zero, the algebraic de Rham cohomology groups (defined using Kähler differentials) are isomorphic to the sheaf cohomology groups of the constant sheaf
• This theorem establishes a connection between the algebraic and analytic theories of cohomology and has important consequences in the study of algebraic cycles and motives
• The proof of the algebraic de Rham theorem involves the construction of the Hodge filtration on the de Rham complex and the comparison with the Hodge-to-de Rham spectral sequence