Sheaf Cohomology
Introduction
Sheaf cohomology is a mathematical tool used in algebraic geometry, topology, and complex analysis to study the global properties of spaces by examining local data. It provides a way to systematically handle the information contained in sheaves, which are structures that associate data to open sets of a topological space. The theory of sheaf cohomology has profound implications in various areas of mathematics, including the study of algebraic varieties, complex manifolds, and topological spaces.
Historical Background
The concept of sheaf cohomology emerged in the mid-20th century as a generalization of classical cohomology theories. It was developed primarily by Jean Leray, Henri Cartan, and Alexander Grothendieck. Leray introduced the notion of sheaves during World War II, and Cartan and his students further developed the theory, leading to the formulation of sheaf cohomology. Grothendieck's work in the 1950s and 1960s significantly advanced the field, particularly through his development of Grothendieck's six operations, which unified various cohomological theories.
Basic Concepts
Sheaves
A sheaf is a mathematical structure that assigns data to the open sets of a topological space in a way that is compatible with the restriction of open sets. Formally, a sheaf \(\mathcal{F}\) on a topological space \(X\) is a functor from the category of open sets of \(X\) (with inclusions as morphisms) to the category of sets (or another category, such as abelian groups or modules), satisfying two conditions:
1. **Locality**: If \(\{U_i\}\) is an open cover of \(U\) and \(s, t \in \mathcal{F}(U)\) such that \(s|_{U_i} = t|_{U_i}\) for all \(i\), then \(s = t\). 2. **Gluing**: If \(\{U_i\}\) is an open cover of \(U\) and \(s_i \in \mathcal{F}(U_i)\) are sections such that \(s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}\) for all \(i, j\), then there exists a section \(s \in \mathcal{F}(U)\) such that \(s|_{U_i} = s_i\) for all \(i\).
Cohomology of Sheaves
The cohomology of a sheaf \(\mathcal{F}\) on a topological space \(X\) is a sequence of abelian groups \(H^n(X, \mathcal{F})\) for \(n \geq 0\). These groups measure the extent to which local sections of \(\mathcal{F}\) fail to glue together to form global sections. The zeroth cohomology group \(H^0(X, \mathcal{F})\) is simply the group of global sections of \(\mathcal{F}\).
The higher cohomology groups \(H^n(X, \mathcal{F})\) for \(n > 0\) are defined using derived functors. Specifically, they are the right derived functors of the global section functor \(\Gamma(X, -)\). This means that \(H^n(X, \mathcal{F})\) is computed by taking an injective resolution of \(\mathcal{F}\) and applying the global section functor to each term of the resolution.
Computation of Sheaf Cohomology
Čech Cohomology
One of the primary methods for computing sheaf cohomology is Čech cohomology. Given an open cover \(\mathcal{U} = \{U_i\}\) of \(X\), the Čech complex associated with \(\mathcal{U}\) and a sheaf \(\mathcal{F}\) is a cochain complex whose cohomology groups are the Čech cohomology groups \(\check{H}^n(\mathcal{U}, \mathcal{F})\). These groups approximate the sheaf cohomology groups \(H^n(X, \mathcal{F})\), and under certain conditions, they are isomorphic.
Spectral Sequences
Spectral sequences are another powerful tool for computing sheaf cohomology. A spectral sequence is a sequence of pages, each consisting of a double complex, that converges to the desired cohomology group. The Leray spectral sequence and the Grothendieck spectral sequence are two important examples used in the computation of sheaf cohomology.
Derived Categories
The language of derived categories provides a modern framework for understanding sheaf cohomology. In this context, the derived category \(D(X)\) of a topological space \(X\) is the category whose objects are complexes of sheaves on \(X\), and whose morphisms are chain maps up to homotopy. The cohomology groups \(H^n(X, \mathcal{F})\) can be interpreted as the Hom groups in the derived category.
Applications
Algebraic Geometry
In algebraic geometry, sheaf cohomology is used to study the properties of algebraic varieties. For example, the cohomology of coherent sheaves provides information about the global sections of line bundles and higher-rank vector bundles. The Riemann-Roch theorem and its generalizations, such as the Hirzebruch-Riemann-Roch theorem and the Grothendieck-Riemann-Roch theorem, are fundamental results that relate the cohomology of sheaves to the geometry of varieties.
Complex Analysis
In complex analysis, sheaf cohomology is used to study complex manifolds. The Dolbeault cohomology groups, which are defined using sheaves of differential forms, play a crucial role in the theory of complex manifolds. The Hodge decomposition theorem, which decomposes the cohomology of a compact Kähler manifold into Dolbeault cohomology groups, is a significant application of sheaf cohomology.
Topology
In topology, sheaf cohomology provides a unifying framework for various classical cohomology theories. For example, the singular cohomology of a topological space can be interpreted as the sheaf cohomology of the constant sheaf. The de Rham cohomology of a smooth manifold is another example that fits into the framework of sheaf cohomology.
Advanced Topics
Derived Functors
Derived functors are a central concept in the theory of sheaf cohomology. Given a left exact functor \(F\) between abelian categories, the right derived functors \(R^nF\) are defined using injective resolutions. In the case of sheaf cohomology, the global section functor \(\Gamma(X, -)\) is left exact, and its right derived functors \(R^n\Gamma(X, -)\) are the sheaf cohomology groups \(H^n(X, -)\).
Grothendieck's Six Operations
Grothendieck's six operations provide a powerful formalism for working with sheaf cohomology in the context of derived categories. The six operations are:
1. **Direct image** \(f_*\) 2. **Inverse image** \(f^*\) 3. **Direct image with proper support** \(f_!\) 4. **Inverse image with compact support** \(f^!\) 5. **Tensor product** \(\otimes\) 6. **Hom** \(\mathcal{H}om\)
These operations satisfy various adjunctions and compatibilities that are crucial for the study of sheaf cohomology.
Duality Theorems
Duality theorems are fundamental results in sheaf cohomology that relate the cohomology of a sheaf to the cohomology of its dual. The Serre duality theorem, for example, provides an isomorphism between the cohomology groups of a coherent sheaf on a smooth projective variety and the cohomology groups of its dual sheaf. The Grothendieck duality theorem generalizes Serre duality to the setting of derived categories and proper morphisms.
Conclusion
Sheaf cohomology is a rich and versatile tool in modern mathematics, with applications spanning algebraic geometry, complex analysis, and topology. Its development has led to deep insights and powerful techniques that continue to influence various areas of mathematical research.