Hochschild Cohomology

Definition

Hochschild cohomology is an important concept in algebraic topology and homological algebra that generalizes the notion of cohomology to algebraic structures such as algebras and categories. It provides a way to study extensions, deformations, and automorphisms of algebras using derived categories. This cohomology theory captures information about the algebra's structure by associating a sequence of abelian groups or vector spaces to it, reflecting its derived properties.

5 Must Know Facts For Your Next Test

1. Hochschild cohomology is typically denoted as HH^*(A) for an algebra A, where * indicates the degree of cohomology.
2. It can be computed using a projective resolution of the algebra, allowing one to use techniques from homological algebra.
3. The zeroth Hochschild cohomology group HH^0(A) corresponds to the center of the algebra A.
4. Hochschild cohomology has applications in various fields such as deformation theory, category theory, and representation theory.
5. The spectral sequence associated with Hochschild cohomology provides tools for computing these cohomology groups in more complex cases.

Review Questions

• How does Hochschild cohomology relate to the concept of derived categories?
  • Hochschild cohomology relates closely to derived categories by utilizing them to compute the cohomological properties of algebras. Specifically, derived categories allow us to handle chain complexes that arise in defining Hochschild cohomology and facilitate working with resolutions. By viewing algebras through the lens of derived categories, we can analyze their extensions and deformations in a structured way, revealing deeper insights into their homotopical behavior.
• Discuss the significance of the zeroth Hochschild cohomology group HH^0(A) and what it represents within the context of an algebra A.
  • The zeroth Hochschild cohomology group HH^0(A) is significant because it identifies the center of the algebra A. This means that it captures elements of A that commute with all other elements. Understanding this group gives insight into the structure of the algebra itself, influencing both its representation theory and its interactions with other mathematical objects. Therefore, HH^0 plays a critical role in bridging algebraic and geometric perspectives.
• Evaluate how Hochschild cohomology can impact our understanding of deformation theory and what implications this has on derived categories.
  • Hochschild cohomology significantly impacts our understanding of deformation theory by providing tools to study how algebras can be 'deformed' or modified while preserving certain structures. The relationship between Hochschild cohomology and derived categories allows us to understand not just individual algebras but families of algebras through their deformations. This connection reveals how geometric concepts manifest in algebraic settings and fosters deeper investigations into the classification and behavior of algebras under deformation, reflecting broader themes in modern mathematics.

Related terms

Derived Category: A derived category is a framework used in homological algebra that allows for the manipulation of chain complexes and their morphisms in a way that focuses on their homotopical properties.

Cohomology: Cohomology is a mathematical tool used to associate a sequence of algebraic objects (like groups or rings) to a topological space, providing insights into its shape and structure.

Gerstenhaber Algebra: A Gerstenhaber algebra is an algebraic structure that combines the operations of cohomology and deformation theory, capturing both the associative and Lie algebraic properties of Hochschild cohomology.