5.1 Čech cohomology and derived functors

Čech cohomology and derived functors are powerful tools for studying sheaves on topological spaces. They provide a way to measure the global properties of sheaves using local data, connecting algebraic and geometric structures.

These concepts are crucial for understanding cohomology theories in algebraic geometry. They allow us to compute important invariants of varieties and sheaves, shedding light on their geometric and topological properties.

Čech cohomology and sheaf cohomology

Definition and construction of Čech cohomology

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• Čech cohomology is a cohomology theory for sheaves on a topological space, based on open covers of the space
• Given a sheaf $F$ and an open cover $U$ of a topological space $X$, the Čech complex $C^{\bullet}(U, F)$ is defined as the complex of $F$-valued functions on intersections of open sets in $U$
• The Čech cohomology groups $\check{H}^i(U, F)$ are defined as the cohomology groups of the Čech complex $C^{\bullet}(U, F)$
• Čech cohomology is invariant under refinement of open covers, leading to the definition of Čech cohomology $\check{H}^i(X, F)$ as the direct limit over all open covers of $X$

Relationship between Čech cohomology and sheaf cohomology

• Čech cohomology is related to sheaf cohomology: if $X$ is a paracompact Hausdorff space and $F$ is a sheaf on $X$, then the Čech cohomology groups $\check{H}^i(X, F)$ are canonically isomorphic to the sheaf cohomology groups $H^i(X, F)$ for all $i \geq 0$
• The Čech-to-derived spectral sequence relates Čech cohomology to derived functor cohomology for sheaves on a topological space
• The Leray spectral sequence for Čech cohomology relates the Čech cohomology of a sheaf on a space $X$ to the Čech cohomology of its direct image sheaf on a subspace $Y$

Čech cohomology computation

Computing Čech cohomology on algebraic varieties

• For a sheaf $F$ on an algebraic variety $X$, the Čech cohomology groups $\check{H}^i(X, F)$ can be computed using an affine open cover of $X$
• On affine varieties, Čech cohomology of quasi-coherent sheaves can be computed using Čech complexes of modules over the coordinate ring
• For a locally free sheaf (vector bundle) on a projective variety, Čech cohomology can be computed using the Serre twisting sheaves $\mathcal{O}(n)$ and the Euler exact sequence

Examples of Čech cohomology computations

• Čech cohomology of the structure sheaf $\mathcal{O}_X$ on a projective space $\mathbb{P}^n$ is isomorphic to the singular cohomology of $\mathbb{P}^n$, with $\check{H}^i(\mathbb{P}^n, \mathcal{O}_X) = \mathbb{C}$ for $i = 0, n$ and $0$ otherwise
• On a smooth projective curve $X$, the Čech cohomology group $\check{H}^1(X, \mathcal{O}_X)$ is isomorphic to the space of global holomorphic differentials on $X$
• For a torus $(\mathbb{C}^{\times})^n$, the Čech cohomology groups $\check{H}^i((\mathbb{C}^{\times})^n, \mathcal{O})$ can be computed using the Koszul complex and are isomorphic to the exterior algebra $\wedge^i \mathbb{C}^n$

Derived functors and cohomology theories

Construction and properties of derived functors

• Derived functors are a way to extend a left or right exact functor between abelian categories to a sequence of functors that measure the failure of exactness
• For a left exact functor $F: \mathcal{A} \to \mathcal{B}$ between abelian categories, the right derived functors $R^iF$ are defined by $R^iF(A) = H^i(F(I^{\bullet}))$, where $I^{\bullet}$ is an injective resolution of the object $A$ in $\mathcal{A}$
• Dually, for a right exact functor $G: \mathcal{A} \to \mathcal{B}$, the left derived functors $L_iG$ are defined using projective resolutions
• Derived functors are unique up to natural isomorphism and independent of the choice of resolution
• The existence and uniqueness of derived functors can be proven, and they form a universal $\delta$-functor extending the original functor

Applications of derived functors to cohomology theories

• Sheaf cohomology can be defined as the right derived functors of the global sections functor $\Gamma(X, -)$ from the category of sheaves on $X$ to the category of abelian groups
• Other cohomology theories, such as group cohomology and Lie algebra cohomology, can also be interpreted as derived functors of suitable functors
• The Grothendieck spectral sequence relates the derived functors of a composition of functors to the derived functors of the individual functors, providing a powerful tool for computing cohomology groups

Properties of Čech cohomology and derived functors

Cohomological properties of Čech cohomology

• Čech cohomology is a cohomological $\delta$-functor, satisfying the long exact sequence of cohomology associated to a short exact sequence of sheaves
• Čech cohomology is invariant under refinement of open covers, allowing for the definition of Čech cohomology $\check{H}^i(X, F)$ as the direct limit over all open covers of $X$
• The Čech-to-derived spectral sequence relates Čech cohomology to derived functor cohomology for sheaves on a topological space, providing a comparison between the two cohomology theories

Spectral sequences involving Čech cohomology and derived functors

• The Leray spectral sequence for Čech cohomology relates the Čech cohomology of a sheaf on a space $X$ to the Čech cohomology of its direct image sheaf on a subspace $Y$, allowing for the computation of cohomology groups on a space using a subspace
• The Grothendieck spectral sequence relates the derived functors of a composition of functors to the derived functors of the individual functors, providing a powerful tool for computing cohomology groups in various settings (derived categories, sheaves on topological spaces, etc.)
• Spectral sequences are a common tool in homological algebra and algebraic geometry, allowing for the computation of cohomology groups by successively approximating the desired groups using simpler ones