Hochschild Homology

Definition

Hochschild homology is an important concept in algebraic topology and noncommutative geometry that measures the homological properties of algebras and modules. It provides a way to study the structure of algebras through chains of their bimodules and has deep connections to Hochschild cohomology, which focuses on cohomological aspects. This duality between homology and cohomology forms a critical part of understanding the algebraic invariants in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. Hochschild homology can be computed using a projective resolution of the algebra in question, which involves the algebra's bimodules.
2. The groups of Hochschild homology are denoted as $$HH_n(A)$$ for an algebra $$A$$ and they depend on both the algebra structure and its module structure.
3. There exists a spectral sequence that relates Hochschild homology to other types of homology, showcasing its fundamental role in algebraic topology.
4. Hochschild homology enjoys functoriality, meaning it behaves well with respect to morphisms of algebras, allowing one to transfer results across related structures.
5. The relationship between Hochschild homology and cohomology is crucial; for example, there is a universal coefficient theorem connecting them.

Review Questions

• How does Hochschild homology relate to the study of bimodules, and why is this relationship significant?
  • Hochschild homology relies heavily on the concept of bimodules because it measures the homological properties of an algebra by examining chains formed from its bimodules. This relationship is significant because it allows mathematicians to derive important invariants from algebras and understand their structure in depth. Bimodules provide a way to connect both sides of an algebra's action, facilitating the computation of homology groups that reflect the underlying algebraic behavior.
• Discuss the importance of functoriality in Hochschild homology and how it influences results when moving between different algebras.
  • Functoriality in Hochschild homology ensures that if there is a morphism between two algebras, then there exists an induced map between their Hochschild homology groups. This property influences results significantly because it allows mathematicians to carry information about the structure and invariants of one algebra over to another. As a result, many results can be generalized or transferred across different contexts, making Hochschild homology a powerful tool in the study of algebraic structures.
• Evaluate how the relationship between Hochschild homology and cohomology enriches our understanding of algebraic structures in noncommutative geometry.
  • The relationship between Hochschild homology and cohomology enriches our understanding by providing dual perspectives on algebraic structures. Homology focuses on chains that measure dimensions while cohomology captures obstructions and extensions, resulting in a complementary framework. This duality enables deeper insights into noncommutative geometry, where traditional geometric intuition might not apply. By exploring both sides, researchers can uncover rich structures within algebras that are not readily visible through either lens alone.

Related terms

Bimodule:

A bimodule is a module that is simultaneously a left module and a right module over two possibly different algebras, allowing interactions between them.

Cohomology: Cohomology is a mathematical tool used to study topological spaces and algebraic structures through the use of cochains, providing insights into their properties.

Homological Algebra: Homological algebra is the branch of mathematics that studies homology in a general algebraic setting, often focusing on complexes of modules or abelian groups.