Hochschild Cohomology

Definition

Hochschild cohomology is a mathematical framework that studies the properties of algebraic structures, particularly associative algebras, by examining their derived functors. It plays a crucial role in understanding the structure of algebras and their modules, providing insights into deformation theory and representation theory. This concept connects to cyclic cohomology through its formulation and applications in noncommutative geometry, as both theories investigate similar algebraic phenomena but from different perspectives.

5 Must Know Facts For Your Next Test

1. Hochschild cohomology groups are defined using chains of multi-linear maps and can be interpreted as measuring how much an algebra fails to be flat over its base field.
2. The zeroth Hochschild cohomology group represents the space of derivations of the algebra, while higher groups capture more complex interactions within the algebra's structure.
3. Hochschild cohomology is particularly useful in the study of noncommutative algebras, where it helps classify and understand their representations and deformations.
4. There is a close relationship between Hochschild cohomology and Hochschild homology; together, they provide a complete picture of the algebra's structure and invariants.
5. One of the key applications of Hochschild cohomology is in quantization problems in physics, where it aids in transitioning from classical to quantum frameworks.

Review Questions

• How does Hochschild cohomology relate to cyclic cohomology in terms of their goals and methods?
  • Hochschild cohomology and cyclic cohomology both aim to understand the properties of algebras, but they do so from different angles. While Hochschild focuses on associative algebras using derived functors and multi-linear maps, cyclic cohomology incorporates cyclicity, looking at how structures behave under cyclic permutations. Both methods explore invariants and can be used to derive information about deformations and representations, making them complementary tools in noncommutative geometry.
• What are the implications of deriving functors for understanding the structure of Hochschild cohomology?
  • Derived functors are essential for understanding Hochschild cohomology because they provide a systematic way to compute cohomological dimensions. In this context, they help define how algebras interact with modules through resolutions, allowing for deeper insights into deformation theory. The derived approach makes it possible to analyze the relationships between different algebraic objects, revealing how they may deform under perturbations or transformations.
• In what ways does Hochschild cohomology contribute to deformation theory and its applications in modern mathematics?
  • Hochschild cohomology significantly contributes to deformation theory by providing tools to classify algebraic structures under small perturbations. It allows mathematicians to analyze moduli spaces and study how certain properties change when algebras are deformed. This interaction between Hochschild cohomology and deformation theory has led to advancements in areas such as quantum mechanics and string theory, highlighting its importance in both pure mathematics and theoretical physics.

Related terms

Cyclic Cohomology: A cohomology theory that extends classical cohomology by including cyclic structures, focusing on the properties of noncommutative algebras and their invariants.

Derived Functors: Functors that arise from homological algebra, used to compute cohomology groups by generalizing the notion of taking derived sequences of projective or injective resolutions.

Deformation Theory: A mathematical study that explores how algebraic structures change when small perturbations are applied, often linked with Hochschild cohomology to understand moduli spaces of algebras.