🧬Cohomology Theory Unit 10 – Sheaf cohomology

Key Concepts and Definitions

• Sheaf cohomology studies the global properties of a topological space by analyzing the local behavior of sheaves on that space
• A sheaf $\mathcal{F}$ on a topological space $X$ consists of a collection of abelian groups $\mathcal{F}(U)$ for each open set $U \subset X$ and restriction maps $\rho_{U,V}: \mathcal{F}(U) \to \mathcal{F}(V)$ for each inclusion $V \subset U$ of open sets
  • These restriction maps satisfy the compatibility condition $\rho_{U,W} = \rho_{V,W} \circ \rho_{U,V}$ for any open sets $W \subset V \subset U$
• The cohomology groups $H^i(X, \mathcal{F})$ measure the obstruction to extending local sections of the sheaf $\mathcal{F}$ to global sections
• The $i$-th cohomology group $H^i(X, \mathcal{F})$ is defined as the $i$-th derived functor of the global sections functor $\Gamma(X, -)$ applied to the sheaf $\mathcal{F}$
• The cohomology groups are invariant under certain operations on sheaves, such as the pushforward and pullback functors
• A presheaf is a collection of abelian groups assigned to open sets of a topological space that satisfies the restriction map compatibility condition but may not satisfy the gluing condition of sheaves
• The sheafification functor associates a unique sheaf to every presheaf by enforcing the gluing condition

Historical Context and Development

• Sheaf cohomology emerged in the 1940s and 1950s as a powerful tool for studying complex analytic spaces and algebraic varieties
• The concept of a sheaf was introduced by Jean Leray in the 1940s as a way to study the topology of fiber bundles and their cohomology
• Henri Cartan and Jean-Pierre Serre further developed sheaf theory and introduced the notion of a coherent sheaf in the context of complex analytic geometry
• Alexander Grothendieck revolutionized algebraic geometry in the 1950s and 1960s by introducing schemes and developing a general theory of sheaves and their cohomology
  • Grothendieck's approach unified various cohomology theories, such as Čech cohomology and de Rham cohomology, under the framework of derived functors
• The Grothendieck topology, a generalization of the Zariski topology, provided a foundation for the study of sheaves on arbitrary sites
• Sheaf cohomology has since found applications in diverse areas of mathematics, including complex analysis, algebraic topology, representation theory, and mathematical physics

Sheaves and Presheaves

• A presheaf $\mathcal{F}$ on a topological space $X$ assigns an abelian group $\mathcal{F}(U)$ to each open set $U \subset X$ and a restriction map $\rho_{U,V}: \mathcal{F}(U) \to \mathcal{F}(V)$ for each inclusion $V \subset U$ of open sets
  • The restriction maps satisfy the compatibility condition $\rho_{U,W} = \rho_{V,W} \circ \rho_{U,V}$ for any open sets $W \subset V \subset U$
• A sheaf is a presheaf that satisfies the gluing condition: if $\{U_i\}$ is an open cover of an open set $U$ and $s_i \in \mathcal{F}(U_i)$ are sections such that $\rho_{U_i, U_i \cap U_j}(s_i) = \rho_{U_j, U_i \cap U_j}(s_j)$ for all $i, j$, then there exists a unique section $s \in \mathcal{F}(U)$ such that $\rho_{U, U_i}(s) = s_i$ for all $i$
• Examples of sheaves include the sheaf of continuous functions, the sheaf of holomorphic functions, and the sheaf of regular functions on a variety
• The sheafification functor $\mathcal{F} \mapsto \mathcal{F}^+$ associates a unique sheaf $\mathcal{F}^+$ to every presheaf $\mathcal{F}$ by enforcing the gluing condition
• Sheaf morphisms are natural transformations between sheaves that commute with the restriction maps
• The category of sheaves on a topological space $X$ is an abelian category, which allows for the use of homological algebra techniques in studying sheaves and their cohomology

Cohomology Groups and Their Construction

• The $i$-th cohomology group $H^i(X, \mathcal{F})$ of a sheaf $\mathcal{F}$ on a topological space $X$ measures the obstruction to extending local sections of $\mathcal{F}$ to global sections
• Cohomology groups are defined using the derived functors of the global sections functor $\Gamma(X, -)$, which assigns to each sheaf $\mathcal{F}$ the abelian group of global sections $\Gamma(X, \mathcal{F})$
  • The $i$-th cohomology group is given by $H^i(X, \mathcal{F}) = R^i\Gamma(X, \mathcal{F})$, where $R^i\Gamma$ denotes the $i$-th right derived functor of $\Gamma$
• To compute the derived functors, one uses an injective resolution of the sheaf $\mathcal{F}$, which is a long exact sequence of injective sheaves $0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots$ that is exact at $\mathcal{F}$
  • The cohomology groups are then computed as the cohomology of the complex $0 \to \Gamma(X, \mathcal{I}^0) \to \Gamma(X, \mathcal{I}^1) \to \cdots$
• The cohomology groups are functorial in both the space $X$ and the sheaf $\mathcal{F}$, meaning that continuous maps between spaces and sheaf morphisms induce homomorphisms between the corresponding cohomology groups
• The long exact sequence in cohomology relates the cohomology groups of a short exact sequence of sheaves $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ via connecting homomorphisms
• The cohomology groups satisfy various properties, such as the existence of cup products, which give them additional algebraic structure

Čech Cohomology

• Čech cohomology is a concrete method for computing the cohomology groups of a sheaf on a topological space using open covers
• Given an open cover $\mathcal{U} = \{U_i\}$ of a topological space $X$ and a sheaf $\mathcal{F}$ on $X$, the Čech complex $\check{C}^\bullet(\mathcal{U}, \mathcal{F})$ is defined as follows:
  • The $n$-th term is given by $\check{C}^n(\mathcal{U}, \mathcal{F}) = \prod_{i_0 < \cdots < i_n} \mathcal{F}(U_{i_0} \cap \cdots \cap U_{i_n})$
  • The differential $\delta^n: \check{C}^n(\mathcal{U}, \mathcal{F}) \to \check{C}^{n+1}(\mathcal{U}, \mathcal{F})$ is defined using the restriction maps and alternating signs
• The $i$-th Čech cohomology group with respect to the cover $\mathcal{U}$ is defined as the $i$-th cohomology group of the Čech complex: $\check{H}^i(\mathcal{U}, \mathcal{F}) = H^i(\check{C}^\bullet(\mathcal{U}, \mathcal{F}))$
• Refining the open cover leads to a homomorphism between the corresponding Čech cohomology groups, and the direct limit of these groups over all open covers is isomorphic to the sheaf cohomology groups: $H^i(X, \mathcal{F}) \cong \varinjlim_\mathcal{U} \check{H}^i(\mathcal{U}, \mathcal{F})$
• Čech cohomology is particularly useful for computing the cohomology of coherent sheaves on complex manifolds, as the Čech complex can be constructed using holomorphic functions
• The Leray spectral sequence relates the Čech cohomology of a sheaf on a space to the sheaf cohomology of its pushforward under a continuous map

Applications in Algebraic Geometry

• Sheaf cohomology is a fundamental tool in algebraic geometry for studying the properties of algebraic varieties and schemes
• The cohomology groups of the structure sheaf $\mathcal{O}_X$ on a variety $X$ provide important invariants, such as the dimension and the arithmetic genus
  • For a smooth projective variety $X$ over a field, the dimension is given by the largest integer $n$ such that $H^n(X, \mathcal{O}_X) \neq 0$
  • The arithmetic genus of a projective variety $X$ is defined as $p_a(X) = (-1)^{\dim X}(\chi(\mathcal{O}_X) - 1)$, where $\chi(\mathcal{O}_X) = \sum_{i=0}^{\dim X} (-1)^i \dim H^i(X, \mathcal{O}_X)$ is the Euler characteristic
• The cohomology of the canonical sheaf $\omega_X$ and its twists $\omega_X(n)$ plays a crucial role in the classification of algebraic varieties, especially in the context of the minimal model program
• Serre duality relates the cohomology groups of a coherent sheaf $\mathcal{F}$ on a smooth projective variety $X$ to the cohomology groups of its dual sheaf tensored with the canonical sheaf: $H^i(X, \mathcal{F}) \cong H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)^\vee$, where $n = \dim X$
• The Riemann-Roch theorem expresses the Euler characteristic of a coherent sheaf on a smooth projective variety in terms of its Chern character and the Todd class of the tangent bundle
• Sheaf cohomology is used to define and study important moduli spaces in algebraic geometry, such as the Picard scheme parametrizing line bundles on a variety and the Hilbert scheme parametrizing closed subschemes

Connections to Other Mathematical Theories

• Sheaf cohomology is closely related to other cohomology theories in mathematics, such as singular cohomology, de Rham cohomology, and étale cohomology
• For a topological space $X$, the singular cohomology groups $H^i(X, \mathbb{Z})$ with integer coefficients can be interpreted as the sheaf cohomology groups of the constant sheaf $\underline{\mathbb{Z}}$ on $X$
• On a smooth manifold $M$, the de Rham cohomology groups $H^i_{dR}(M)$ are isomorphic to the sheaf cohomology groups of the sheaf of smooth differential forms $\Omega^i_M$
  • This isomorphism is a consequence of the Poincaré lemma, which states that the sheaf of closed differential forms is a resolution of the constant sheaf $\underline{\mathbb{R}}$
• Étale cohomology, introduced by Grothendieck, is a cohomology theory for schemes that takes into account the arithmetic properties of the base field
  • Étale cohomology groups are defined using sheaves on the étale site of a scheme, which is a Grothendieck topology that captures the local structure of the scheme in the étale topology
• Sheaf cohomology has applications in complex analysis, where it is used to study the cohomology of holomorphic sheaves on complex manifolds and to prove vanishing theorems, such as the Kodaira vanishing theorem
• In representation theory, sheaf cohomology is used to construct and study representations of algebraic groups and Lie algebras, such as the Borel-Weil-Bott theorem, which relates the cohomology of line bundles on flag varieties to irreducible representations

Problem-Solving Techniques and Examples

• When computing sheaf cohomology, it is often useful to employ spectral sequences, which are algebraic tools that relate the cohomology of a complex of sheaves to the cohomology of the individual sheaves
  • The Leray spectral sequence is particularly useful for computing the sheaf cohomology of a pushforward sheaf
• In some cases, sheaf cohomology can be computed using a Čech cover and the Čech complex, especially when working with coherent sheaves on complex manifolds
  • For example, to compute the cohomology of the structure sheaf on the complex projective space $\mathbb{P}^n$, one can use the standard affine cover and the Čech complex of holomorphic functions
• Vanishing theorems, such as the Kodaira vanishing theorem and the Nakano vanishing theorem, provide powerful tools for determining the vanishing of certain cohomology groups
  • These theorems often rely on the positivity properties of the canonical bundle or the curvature of a hermitian metric on a vector bundle
• The splitting principle is a technique for reducing computations involving vector bundles to computations involving line bundles, which are often easier to handle
  • This principle states that every vector bundle can be pulled back to a sum of line bundles under a suitable morphism, and the cohomology of the original bundle can be recovered from the cohomology of the line bundles
• The snake lemma is a useful tool for relating the cohomology groups of sheaves in a short exact sequence, especially when combined with the long exact sequence in cohomology
  • For example, if $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ is a short exact sequence of sheaves and the cohomology of two of the sheaves is known, the snake lemma can be used to determine the cohomology of the third sheaf
• When working with algebraic varieties, it is often helpful to consider the cohomology of the structure sheaf and its twists, as well as the cohomology of the canonical sheaf and its powers
  • The Riemann-Roch theorem and Serre duality can be used to relate the cohomology of these sheaves and to compute important invariants, such as the arithmetic genus and the plurigenera