Hochschild Homology

Definition

Hochschild homology is a powerful tool in homological algebra that provides a way to study algebras by analyzing their structures and relationships through a chain complex. It captures information about the relationships between an algebra and its modules, offering insights into their properties and interactions. This concept also plays a significant role in derived categories and has connections to other areas like cyclic homology and deformation theory.

5 Must Know Facts For Your Next Test

1. Hochschild homology is defined for associative algebras over a field and can be computed using the bar resolution.
2. The n-th Hochschild homology group of an algebra A, denoted by HH_n(A), measures the extent to which A fails to be projective as a bimodule over itself.
3. Hochschild homology is particularly useful in characterizing the structure of a given algebra and understanding its module category.
4. One important property is that HH_0(A) is isomorphic to the center of the algebra A, which captures central elements that commute with all other elements.
5. There is a spectral sequence associated with Hochschild homology that helps relate it to cyclic homology, revealing deep connections between these two theories.

Review Questions

• How does Hochschild homology relate to the study of associative algebras and their modules?
  • Hochschild homology provides a framework for studying associative algebras by examining their structure through chain complexes. Specifically, it allows for the investigation of how an algebra interacts with its modules via the computation of homology groups. These groups offer insights into projective properties and reveal important information about central elements within the algebra, deepening our understanding of the algebra's structure.
• Discuss the importance of the bar resolution in computing Hochschild homology for an algebra A.
  • The bar resolution is crucial for calculating Hochschild homology because it serves as a projective resolution of the algebra A when viewed as a bimodule over itself. By applying this resolution, one constructs a chain complex that allows us to compute the n-th Hochschild homology groups. This method not only facilitates calculations but also highlights how the structure of A influences its homological properties, providing valuable insights into its module category.
• Evaluate the significance of connecting Hochschild homology with cyclic homology, especially in terms of their spectral sequences.
  • The connection between Hochschild and cyclic homology is significant because it highlights fundamental relationships within noncommutative geometry. The spectral sequence associated with Hochschild homology can relate its computations to those of cyclic homology, demonstrating how these two areas intersect. This relationship enriches our understanding of both theories by providing tools for analyzing algebras that are more complex than what each can handle alone, thus broadening our insight into their respective structures and behaviors.

Related terms

Cyclic Homology: A variant of Hochschild homology that incorporates the notion of cyclic symmetry, focusing on the study of noncommutative spaces and their structures.

Chain Complex: A sequence of abelian groups or modules connected by homomorphisms, allowing the calculation of homology and cohomology groups.

Derived Category: A category that captures the derived functors of a given category, providing a framework for studying homological properties and complexes.