5.2 Čech cohomology

Čech cohomology is a powerful tool for studying sheaves on topological spaces. It uses open covers to analyze global properties through local behavior, bridging the gap between local and global perspectives in mathematics.

This approach connects to broader themes in algebraic topology and geometry. By examining how local data pieces together, Čech cohomology provides insights into the structure of spaces and sheaves, forming a key link between algebra and topology.

Čech cohomology definition

• Čech cohomology is a cohomology theory for sheaves on a topological space, introduced by Eduard Čech in the 1930s
• It is based on the idea of covering a space with open sets and studying the global properties of a sheaf by looking at its local behavior on the open sets

Presheaves vs sheaves

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• A presheaf $\mathcal{F}$ on a topological space $X$ assigns to each open set $U \subset X$ an abelian group $\mathcal{F}(U)$ and to each inclusion $V \subset U$ a restriction map $\mathcal{F}(U) \to \mathcal{F}(V)$, satisfying certain compatibility conditions
• A sheaf is a presheaf that satisfies an additional gluing condition: if $\{U_i\}$ is an open cover of $U$ and $s_i \in \mathcal{F}(U_i)$ are sections that agree on overlaps, then there exists a unique section $s \in \mathcal{F}(U)$ restricting to each $s_i$
• Examples of sheaves include the sheaf of continuous functions, the sheaf of smooth functions, and the sheaf of holomorphic functions on a complex manifold

Čech cohomology of presheaves

• Given a presheaf $\mathcal{F}$ and an open cover $\mathcal{U} = \{U_i\}$ of $X$, the Čech complex $\check{C}^\bullet(\mathcal{U}, \mathcal{F})$ is defined as the direct sum of the groups $\mathcal{F}(U_{i_0} \cap \cdots \cap U_{i_p})$ over all $(p+1)$-tuples of indices, with differential given by the alternating sum of restriction maps
• The Čech cohomology groups $\check{H}^p(\mathcal{U}, \mathcal{F})$ are the cohomology groups of this complex
• These groups depend on the choice of cover $\mathcal{U}$, but for a sufficiently fine cover (a good cover), they are isomorphic to the sheaf cohomology groups $H^p(X, \mathcal{F})$

Čech cohomology of sheaves

• For a sheaf $\mathcal{F}$, the Čech cohomology groups $\check{H}^p(\mathcal{U}, \mathcal{F})$ are the same as for the underlying presheaf
• The key difference is that for a sheaf, these groups are independent of the choice of good cover $\mathcal{U}$, and can be denoted simply as $\check{H}^p(X, \mathcal{F})$
• This is because the sheaf condition ensures that the Čech complex is exact for a sufficiently fine cover, and so its cohomology computes the derived functors of the global sections functor

Čech cohomology vs sheaf cohomology

• For a sheaf $\mathcal{F}$ on a topological space $X$, the Čech cohomology groups $\check{H}^p(X, \mathcal{F})$ are isomorphic to the sheaf cohomology groups $H^p(X, \mathcal{F})$, defined as the derived functors of the global sections functor $\Gamma(X, -)$
• This isomorphism is a key result in sheaf theory, known as the Čech-to-derived functor spectral sequence
• One advantage of Čech cohomology is that it is often easier to compute explicitly, using a good cover and the Čech complex
• On the other hand, sheaf cohomology has better functorial properties and is more widely applicable (e.g. to étale cohomology in algebraic geometry)

Čech cohomology computation

• Computing Čech cohomology involves choosing a good cover of the space, constructing the Čech complex, and calculating its cohomology groups
• This process can be simplified using spectral sequences and other algebraic tools

Good covers

• A good cover of a topological space $X$ is an open cover $\mathcal{U} = \{U_i\}$ such that all nonempty finite intersections $U_{i_0} \cap \cdots \cap U_{i_p}$ are contractible (or more generally, acyclic for the sheaf $\mathcal{F}$)
• Examples of spaces admitting good covers include manifolds, CW complexes, and algebraic varieties
• On a space with a good cover, Čech cohomology computes the correct sheaf cohomology groups
• The existence of good covers is a subtle question in general, related to the notion of cohomological dimension

Refinement of 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 are useful because they give rise to maps between Čech complexes, inducing homomorphisms on Čech cohomology
• These refinement maps are isomorphisms for sufficiently fine covers, showing that Čech cohomology is independent of the choice of good cover
• The Leray spectral sequence relates the Čech cohomology for different covers

Čech complex

• The Čech complex $\check{C}^\bullet(\mathcal{U}, \mathcal{F})$ is a cochain complex associated to an open cover $\mathcal{U} = \{U_i\}$ and a sheaf $\mathcal{F}$
• Its $p$-th group is the direct sum of the groups $\mathcal{F}(U_{i_0} \cap \cdots \cap U_{i_p})$ over all $(p+1)$-tuples of indices
• The differential $d^p$ is defined as the alternating sum of restriction maps, with appropriate signs
• The Čech cohomology groups are the cohomology groups of this complex
• For a good cover, the Čech complex is exact and computes the sheaf cohomology

Čech-to-derived functor spectral sequence

• The Čech-to-derived functor spectral sequence is a spectral sequence relating Čech cohomology to sheaf cohomology
• It starts with the Čech complex $\check{C}^\bullet(\mathcal{U}, \mathcal{F})$ and converges to the sheaf cohomology groups $H^p(X, \mathcal{F})$
• The $E_2$ page of the spectral sequence consists of the Čech cohomology groups $\check{H}^p(\mathcal{U}, \mathcal{H}^q(\mathcal{F}))$, where $\mathcal{H}^q(\mathcal{F})$ is the presheaf of $q$-th cohomology groups of $\mathcal{F}$
• For a good cover, this spectral sequence degenerates at the $E_2$ page, giving the isomorphism between Čech and sheaf cohomology
• More generally, the spectral sequence can be used to compute sheaf cohomology in terms of Čech cohomology of simpler sheaves

Applications of Čech cohomology

• Čech cohomology is a powerful tool in algebraic geometry, complex analysis, and other fields
• It is particularly useful for studying coherent sheaves on projective varieties and complex manifolds
• Some key applications include the classification of line bundles, the Serre duality theorem, and the GAGA principle

Coherent sheaves on projective space

• A coherent sheaf on a projective space $\mathbb{P}^n$ is a sheaf of modules over the structure sheaf $\mathcal{O}_{\mathbb{P}^n}$ that is locally presentable as the cokernel of a map between free sheaves
• Examples include the structure sheaf itself, the sheaf of differential forms, and the sheaf of sections of a vector bundle
• The Čech cohomology groups $H^p(\mathbb{P}^n, \mathcal{F})$ of a coherent sheaf $\mathcal{F}$ are finite-dimensional vector spaces over the base field
• These groups can be computed using a standard affine cover of $\mathbb{P}^n$ and the Čech complex

Line bundles and divisors

• A line bundle on a variety $X$ is a locally free sheaf of rank one, i.e. a sheaf that is locally isomorphic to the structure sheaf $\mathcal{O}_X$
• Line bundles are classified by the Picard group $\operatorname{Pic}(X)$, which is isomorphic to the Čech cohomology group $H^1(X, \mathcal{O}_X^*)$ of the sheaf of invertible functions
• A divisor on $X$ is a formal linear combination of codimension-one subvarieties, with integer coefficients
• The group of divisors modulo linear equivalence is isomorphic to $\operatorname{Pic}(X)$, via the correspondence between divisors and line bundles
• The Čech cohomology group $H^0(X, \mathcal{O}(D))$ is the space of global sections of the line bundle associated to a divisor $D$

Serre duality

• Serre duality is a fundamental duality theorem in algebraic geometry, relating the cohomology of a coherent sheaf $\mathcal{F}$ on a projective variety $X$ to the cohomology of its dual sheaf $\mathcal{F}^\vee = \mathcal{H}om(\mathcal{F}, \omega_X)$, where $\omega_X$ is the canonical sheaf
• Specifically, there are natural isomorphisms $H^p(X, \mathcal{F}) \cong H^{n-p}(X, \mathcal{F}^\vee \otimes \omega_X)^\vee$, where $n$ is the dimension of $X$
• This duality can be proved using Čech cohomology and a dualizing complex, or via the more general Grothendieck duality theory
• Serre duality has many applications, such as the Riemann-Roch theorem, the adjunction formula, and the existence of canonical embeddings

GAGA principle

• The GAGA principle, named after Serre's paper "Géométrie Algébrique et Géométrie Analytique", is a comparison theorem between algebraic geometry and complex analytic geometry
• It states that for a complex projective variety $X$, the categories of algebraic coherent sheaves and analytic coherent sheaves are equivalent, and their Čech cohomology groups are isomorphic
• This allows the use of analytic methods (such as sheaf cohomology and Hodge theory) to study algebraic varieties
• The GAGA principle has been generalized to other contexts, such as proper algebraic spaces and Deligne-Mumford stacks

Relation to other cohomology theories

• Čech cohomology is one of several cohomology theories that can be defined for sheaves on a topological space or a variety
• These theories are often related by comparison theorems and spectral sequences
• Understanding these relationships is crucial for applications in algebraic topology, algebraic geometry, and complex analysis

Singular cohomology

• Singular cohomology is a cohomology theory for topological spaces, defined using singular cochains (dual to singular chains)
• For a locally contractible space $X$ and a constant sheaf $\underline{A}$ with coefficients in an abelian group $A$, the Čech cohomology groups $\check{H}^p(X, \underline{A})$ are isomorphic to the singular cohomology groups $H^p(X, A)$
• This isomorphism can be proved using a good cover and the Čech-to-derived functor spectral sequence
• Singular cohomology is a key tool in algebraic topology, used to define invariants such as the Betti numbers and the cup product

de Rham cohomology

• de Rham cohomology is a cohomology theory for smooth manifolds, defined using differential forms
• For a smooth manifold $X$, the de Rham cohomology groups $H^p_{dR}(X)$ are isomorphic to the Čech cohomology groups $\check{H}^p(X, \mathbb{R})$ with coefficients in the constant sheaf $\underline{\mathbb{R}}$
• This isomorphism, known as the de Rham theorem, can be proved using a good cover and the Poincaré lemma (which states that the de Rham complex is locally exact)
• de Rham cohomology is a bridge between topology, analysis, and differential geometry, with applications such as the Hodge theorem and the Atiyah-Singer index theorem

Dolbeault cohomology

• Dolbeault cohomology is a cohomology theory for complex manifolds, defined using $(p,q)$-forms and the Dolbeault operators $\bar{\partial}$
• For a complex manifold $X$, the Dolbeault cohomology groups $H^{p,q}(X)$ are isomorphic to the Čech cohomology groups $\check{H}^q(X, \Omega^p)$ of the sheaf of holomorphic $p$-forms
• This isomorphism can be proved using a good cover and the Dolbeault lemma (which states that the Dolbeault complex is locally exact)
• Dolbeault cohomology is a key tool in complex geometry and analysis, used to study the Hodge decomposition, the $\bar{\partial}$-equation, and the deformation theory of complex structures

Étale cohomology

• Étale cohomology is a cohomology theory for algebraic varieties and schemes, defined using étale sheaves and étale covers
• For a variety $X$ over a separably closed field, the étale cohomology groups $H^p_{ét}(X, \mathbb{Z}/n\mathbb{Z})$ with coefficients in the constant sheaf $\underline{\mathbb{Z}/n\mathbb{Z}}$ are related to the Čech cohomology groups by a spectral sequence
• This spectral sequence, known as the Leray spectral sequence for the étale cover, can be used to compute étale cohomology in terms of Čech cohomology of simpler schemes
• Étale cohomology is a central tool in modern algebraic geometry, with applications such as the Weil conjectures, the étale fundamental group, and the theory of motives

Limitations and extensions

• While Čech cohomology is a powerful and computable theory, it has some limitations and can be generalized in various ways
• These extensions are motivated by questions in algebraic topology, algebraic geometry, and other areas

Cohomological dimension

• The cohomological dimension of a topological space $X$ is the smallest integer $n$ such that $H^p(X, \mathcal{F}) = 0$ for all $p > n$ and all sheaves $\mathcal{F}$
• For a space with finite cohomological dimension, Čech cohomology can be computed using covers of size at most $n+1$
• However, there are spaces (such as the long line) that have infinite cohomological dimension, and for which Čech cohomology is not well-behaved
• The cohomological dimension is related to other notions such as the covering dimension and the Lebesgue covering dimension

Leray spectral sequence

• The Leray spectral sequence is a spectral sequence that relates the Čech cohomology of a sheaf on a space $X$ to the Čech cohomology of its pushforward on another space $Y$, along a continuous map $f: X \to Y$
• It has the form $E_2^{pq} = \check{H}^p(Y, R^q f_* \mathcal{F}) \Rightarrow H^{p+q}(X, \mathcal{F})$, where $R^q f_* \mathcal{F}$ is the higher direct image sheaf
• The Leray spectral sequence can be used to compute cohomology of sheaves on a space in terms of cohomology of simpler sheaves on a simpler space
• Special cases include the Čech-to-derived functor spectral sequence (for the identity map) and the Leray spectral sequence for an étale cover

Hypercoverings and generalizations

• A hypercovering of a topological space $X$ is a generalization of a Čech cover, allowing for higher-dimensional