Mathematics > Algebraic Geometry > Cohomology
Description:
Cohomology is a central concept within algebraic geometry, a branch of mathematics that studies algebraic varieties—geometric manifestations of solutions to systems of polynomial equations. Cohomology provides powerful tools to probe the properties of these varieties by assigning algebraic invariants that encapsulate their topological and geometric features.
In more technical terms, cohomology theories—such as sheaf cohomology, de Rham cohomology, and Čech cohomology—allow mathematicians to systematically study the structure of spaces. Specifically, in algebraic geometry, sheaf cohomology is particularly essential. It uses sheaves, which are algebraic data structures that assign algebraic objects (like groups or rings) to the open sets of a topological space in a way that respects the gluing of these sets, to provide insight into the distribution and interaction of functions, differential forms, and other algebraic entities across an algebraic variety.
Key Concepts:
1. Sheaves and Sheaf Cohomology:
- Sheaf: A sheaf \(\\mathcal{F}\) on a topological space \(X\) assigns to each open set \(U \subseteq X\) an algebraic structure \(\mathcal{F}(U)\) (e.g., an abelian group, ring, etc.). It also provides restriction maps compatible with the inclusion of open sets.
- Cohomology Groups: For a sheaf \(\mathcal{F}\), the cohomology groups \(H^i(X, \mathcal{F})\) are defined. These groups measure the “obstructions” to solving certain local-to-global problems. For example, \(H^0(X, \mathcal{F})\) often represents global sections—elements of \(\mathcal{F}(U)\) when \(U = X\).
2. Čech Cohomology:
- Čech cohomology is a computational approach to cohomology. Given an open cover \(\{U_i\}\) of \(X\) and a sheaf \(\mathcal{F}\) on \(X\), one constructs cohomology groups \( \check{H}^i(\{U_i\}, \mathcal{F}) \) from the intersections of these open sets, paving the way for practical computations of \(H^i(X, \mathcal{F})\) through combinatorial data.
3. De Rham Cohomology:
- This type pertains to smooth manifolds, involving differential forms. The de Rham cohomology groups \( H^i_{\text{dR}}(X) \) are formed by closed forms modulo exact forms, giving a bridge between the algebraic and differential geometric realms.
Mathematical Framework:
For a sheaf \(\mathcal{F}\) on a topological space \(X\), the derived functors of the global section functor \(\Gamma(X, -)\) give rise to the cohomology groups \(H^i(X, \mathcal{F})\). This can be formally expressed as:
\[ H^i(X, \mathcal{F}) = R^i \Gamma(X, \mathcal{F}) \]
where \(R^i\) denotes the \(i\)-th right derived functor.
Understanding these cohomology groups can be visualized through exact sequences. For instance, if we have an exact sequence of sheaves
\[ 0 \to \mathcal{F}’ \to \mathcal{F} \to \mathcal{F}’’ \to 0 \]
it induces a long exact sequence in cohomology:
\[ 0 \to H^0(X, \mathcal{F}‘) \to H^0(X, \mathcal{F}) \to H^0(X, \mathcal{F}’‘) \to H^1(X, \mathcal{F}’) \to \cdots \]
Applications:
Cohomology in algebraic geometry has a wide array of applications, including:
- Classification of Varieties: Understanding the structure and classification of algebraic varieties.
- Intersection Theory: Studying how subvarieties intersect within a given variety.
- Comparison with Topological Methods: Relating algebraic properties of varieties to their topological invariants, fostering a deeper comprehension of their geometric nature.
In essence, cohomology in algebraic geometry serves as an indispensable tool for unraveling the intricate relationships defining the shape and form of algebraic varieties, revealing deep and sometimes surprising connections between abstract algebraic concepts and concrete geometric intuition.