4.3 Sheaf cohomology

Sheaf cohomology is a powerful tool in algebraic geometry that measures how local information combines into global structures. It associates cohomology groups to sheaves on topological spaces, capturing obstructions to solving equations or extending local sections globally.

This approach provides deep insights into the properties of algebraic varieties and their sheaves. By studying coherent sheaves, applying duality theorems, and analyzing projective varieties, sheaf cohomology reveals fundamental geometric and topological features of mathematical objects.

Definition of sheaf cohomology

• Sheaf cohomology is a powerful tool in algebraic geometry and complex analysis that associates cohomology groups to a sheaf on a topological space
• It captures global information about the sheaf and the underlying space by studying the obstructions to solving certain equations or extending local sections globally
• Sheaf cohomology provides a way to measure the deviation of a sheaf from being flasque or acyclic, which are properties that allow for easy computation of global sections

Cohomology groups for sheaves

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• For a sheaf $\mathcal{F}$ on a topological space $X$, the cohomology groups $H^i(X, \mathcal{F})$ are defined as the derived functors of the global section functor $\Gamma(X, -)$
  • The $i$-th cohomology group $H^i(X, \mathcal{F})$ measures the obstruction to solving the sheaf cohomology problem in degree $i$
• The 0-th cohomology group $H^0(X, \mathcal{F})$ is isomorphic to the global sections of the sheaf $\mathcal{F}$
• Higher cohomology groups $H^i(X, \mathcal{F})$ for $i > 0$ encode information about the non-triviality of the sheaf and the complexity of the space $X$

Čech cohomology vs derived functor cohomology

• Čech cohomology is a concrete approach to computing sheaf cohomology using open covers of the space $X$ and the associated Čech complex
  • It is based on the idea of gluing local sections on intersections of open sets
• Derived functor cohomology is a more abstract approach that defines sheaf cohomology as the derived functors of the global section functor
  • It relies on the machinery of homological algebra and provides a more conceptual understanding of sheaf cohomology
• The two approaches are equivalent for a wide class of spaces (paracompact Hausdorff spaces) and sheaves, but derived functor cohomology is more general and applicable in various settings

Computation of sheaf cohomology

• Computing sheaf cohomology is a central problem in algebraic geometry and complex analysis, as it provides valuable information about the sheaf and the underlying space
• There are two main approaches to computing sheaf cohomology: Čech cohomology and the derived functor approach
• The choice of the computational method depends on the specific problem and the properties of the space and the sheaf

Čech cohomology computations

• Čech cohomology is computed using an open cover $\mathcal{U} = \{U_i\}$ of the space $X$ and the associated Čech complex $\check{C}^\bullet(\mathcal{U}, \mathcal{F})$
  • The Čech complex is a cosimplicial object that encodes the local sections of the sheaf on intersections of open sets
• The cohomology groups are obtained by taking the cohomology of the Čech complex, i.e., $\check{H}^i(X, \mathcal{F}) = H^i(\check{C}^\bullet(\mathcal{U}, \mathcal{F}))$
• Čech cohomology is particularly useful for computing sheaf cohomology on spaces with a simple topology (contractible spaces, Stein manifolds) and for acyclic sheaves

Derived functor approach

• The derived functor approach computes sheaf cohomology by resolving the sheaf $\mathcal{F}$ with an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ and applying the global section functor
  • The cohomology groups are obtained as the cohomology of the complex of global sections, i.e., $H^i(X, \mathcal{F}) = H^i(\Gamma(X, \mathcal{I}^\bullet))$
• This approach is more general and applicable in various settings, including algebraic geometry and complex analysis
• The derived functor approach is particularly useful for studying the functorial properties of sheaf cohomology and for proving general theorems

Comparison of computational methods

• Čech cohomology and derived functor cohomology agree for a wide class of spaces (paracompact Hausdorff spaces) and sheaves
  • In these cases, the choice of the computational method is a matter of convenience and the specific problem at hand
• Čech cohomology is often easier to compute explicitly, especially for spaces with a simple topology and acyclic sheaves
• The derived functor approach provides a more conceptual understanding of sheaf cohomology and is more suitable for studying functorial properties and proving general theorems

Applications of sheaf cohomology

• Sheaf cohomology has numerous applications in algebraic geometry, complex analysis, and topology, as it provides a powerful tool for studying geometric and topological properties of spaces and sheaves
• Some of the most notable applications include the classification of vector bundles, Hodge theory for complex manifolds, and the study of the cohomological dimension of topological spaces

Classification of vector bundles

• Sheaf cohomology plays a crucial role in the classification of vector bundles over a topological space or a manifold
• The first Chern class of a line bundle can be interpreted as an element of the second cohomology group $H^2(X, \mathcal{O}_X^\times)$, where $\mathcal{O}_X^\times$ is the sheaf of invertible holomorphic functions
• Higher-rank vector bundles can be studied using the cohomology of the sheaf of endomorphisms and the Atiyah class, which measures the obstruction to the existence of a global connection

Hodge theory for complex manifolds

• Sheaf cohomology is a fundamental tool in the study of Hodge theory for complex manifolds
• The Dolbeault cohomology groups $H^{p,q}(X, \mathcal{O}_X)$, which are computed using the Dolbeault complex of sheaves of holomorphic forms, provide a decomposition of the de Rham cohomology of a complex manifold
• The Hodge decomposition theorem relates the Dolbeault cohomology groups to the sheaf cohomology of the sheaf of holomorphic forms and the constant sheaf, providing a deep connection between complex analysis and algebraic topology

Cohomological dimension of topological spaces

• Sheaf cohomology can be used to define and study the cohomological dimension of a topological space $X$
• The cohomological dimension of $X$ with respect to a sheaf $\mathcal{F}$ is the smallest integer $n$ such that $H^i(X, \mathcal{F}) = 0$ for all $i > n$ and all abelian sheaves $\mathcal{F}$
• The cohomological dimension provides a measure of the complexity of the space and has applications in various areas of mathematics, including algebraic topology and homological algebra

Sheaf cohomology in algebraic geometry

• Sheaf cohomology is an indispensable tool in algebraic geometry, where it is used to study the properties of algebraic varieties and their sheaves
• Some of the most important applications of sheaf cohomology in algebraic geometry include the study of coherent sheaves, the Serre duality theorem, and the relation between cohomology and dimension of projective varieties

Coherent sheaves on algebraic varieties

• Coherent sheaves are a special class of sheaves that arise naturally in algebraic geometry as the sheaves of sections of algebraic vector bundles or as the sheaves of algebraic functions on a variety
• The cohomology of coherent sheaves provides important information about the geometry of the underlying variety and the properties of the sheaf itself
• Vanishing theorems for the cohomology of coherent sheaves, such as the Kodaira vanishing theorem and the Nakano vanishing theorem, play a crucial role in the classification of algebraic varieties

Serre duality theorem

• The Serre duality theorem is a fundamental result in algebraic geometry that relates the cohomology of a coherent sheaf on a projective variety to the cohomology of its dual sheaf
• For a smooth projective variety $X$ of dimension $n$ and a coherent sheaf $\mathcal{F}$ on $X$, the Serre duality theorem states that there is a natural isomorphism $H^i(X, \mathcal{F}) \cong H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)^\vee$, where $\omega_X$ is the canonical sheaf of $X$
• The Serre duality theorem has numerous applications in the study of algebraic curves, surfaces, and higher-dimensional varieties, and it is a key ingredient in the proof of the Riemann-Roch theorem for surfaces

Cohomology and dimension of projective varieties

• Sheaf cohomology can be used to study the dimension and other geometric properties of projective varieties
• For a projective variety $X$ and a coherent sheaf $\mathcal{F}$ on $X$, the dimension of $X$ can be characterized as the smallest integer $n$ such that $H^i(X, \mathcal{F}(k)) = 0$ for all $i > n$ and all sufficiently large $k$, where $\mathcal{F}(k)$ denotes the twist of $\mathcal{F}$ by the $k$-th power of the hyperplane line bundle
• The cohomology of the structure sheaf $\mathcal{O}_X$ and its twists encodes important information about the geometry of the variety, such as its degree, genus, and arithmetic properties

Relation to other cohomology theories

• Sheaf cohomology is closely related to other cohomology theories in mathematics, such as de Rham cohomology, singular cohomology, and the theory of universal δ-functors
• Understanding the connections between these cohomology theories provides a deeper insight into the nature of sheaf cohomology and its place in the broader context of algebraic topology and homological algebra

De Rham cohomology vs sheaf cohomology

• De Rham cohomology is a cohomology theory for smooth manifolds that is defined using the complex of differential forms and the exterior derivative
• For a smooth manifold $X$, the de Rham cohomology groups $H^i_{dR}(X)$ are isomorphic to the sheaf cohomology groups $H^i(X, \mathbb{R}_X)$, where $\mathbb{R}_X$ is the constant sheaf with value $\mathbb{R}$
• This isomorphism, known as the de Rham theorem, provides a bridge between the analytic and topological aspects of cohomology and has important applications in the study of characteristic classes and the topology of manifolds

Singular cohomology vs sheaf cohomology

• Singular cohomology is a cohomology theory for topological spaces that is defined using the complex of singular cochains and the coboundary operator
• For a topological space $X$ and an abelian group $A$, the singular cohomology groups $H^i(X, A)$ are related to the sheaf cohomology groups $H^i(X, A_X)$, where $A_X$ is the constant sheaf with value $A$
• The relation between singular cohomology and sheaf cohomology is given by the universal coefficient theorem, which expresses the singular cohomology groups in terms of the sheaf cohomology groups and the Ext functor

Sheaf cohomology as a universal δ-functor

• Sheaf cohomology can be characterized as a universal δ-functor, which is a sequence of functors that satisfies certain axioms and is universal among such sequences
• The theory of universal δ-functors provides a unified framework for studying various cohomology theories, including sheaf cohomology, group cohomology, and Ext functors
• The universality of sheaf cohomology as a δ-functor has important consequences for the functorial properties of sheaf cohomology and its relation to other cohomology theories

Advanced topics in sheaf cohomology

• Sheaf cohomology is a vast and active area of research, with numerous advanced topics and applications in various branches of mathematics
• Some of the most important advanced topics in sheaf cohomology include hypercohomology and spectral sequences, Grothendieck's algebraic de Rham theorem, and étale cohomology and l-adic sheaves

Hypercohomology and spectral sequences

• Hypercohomology is an extension of sheaf cohomology that allows for the study of complexes of sheaves and their cohomology
• For a complex of sheaves $\mathcal{F}^\bullet$ on a topological space $X$, the hypercohomology groups $\mathbb{H}^i(X, \mathcal{F}^\bullet)$ are defined as the derived functors of the global section functor applied to the complex
• Spectral sequences are a powerful tool in homological algebra that can be used to compute hypercohomology groups and to study the relations between different cohomology theories
• The Leray spectral sequence and the hypercohomology spectral sequence are two important examples of spectral sequences that arise in the study of sheaf cohomology

Grothendieck's algebraic de Rham theorem

• Grothendieck's algebraic de Rham theorem is a far-reaching generalization of the classical de Rham theorem that relates the algebraic de Rham cohomology of a smooth algebraic variety to its sheaf cohomology
• For a smooth algebraic variety $X$ over a field $k$ of characteristic zero, the algebraic de Rham theorem states that there is a natural isomorphism between the algebraic de Rham cohomology groups $H^i_{dR}(X/k)$ and the sheaf cohomology groups $H^i(X, \Omega^\bullet_{X/k})$, where $\Omega^\bullet_{X/k}$ is the complex of sheaves of algebraic differential forms
• The algebraic de Rham theorem has important applications in the study of the topology of algebraic varieties and the theory of motives

Étale cohomology and l-adic sheaves

• Étale cohomology is a cohomology theory for algebraic varieties that is defined using the étale topology and the theory of l-adic sheaves
• For an algebraic variety $X$ over a field $k$ and a prime number $l$ different from the characteristic of $k$, the étale cohomology groups $H^i_{\acute{e}t}(X, \mathbb{Z}/l^n\mathbb{Z})$ are defined as the sheaf cohomology groups of the constant sheaf $\mathbb{Z}/l^n\mathbb{Z}$ in the étale topology
• The l-adic cohomology groups are obtained by taking the inverse limit of the étale cohomology groups over all powers of $l$, and they provide a powerful tool for studying the arithmetic and geometric properties of algebraic varieties
• Étale cohomology and l-adic sheaves have important applications in number theory, algebraic geometry, and the theory of motives, and they play a central role in the proof of the Weil conjectures