Hochschild Complex

Definition

The Hochschild complex is a construction in homological algebra that provides a way to study the properties of associative algebras through their derived categories. It plays a crucial role in connecting algebraic structures with topological and geometric concepts, specifically through the use of derived functors and cohomology theories, including Hochschild homology and cyclic homology.

5 Must Know Facts For Your Next Test

1. The Hochschild complex is built from the tensor products of an algebra with itself, allowing for an analysis of its homological properties.
2. Hochschild homology is defined using the Hochschild complex, which is used to compute invariants that measure the 'size' and 'shape' of an algebra's structure.
3. One significant result involving the Hochschild complex is that it leads to a spectral sequence converging to the Hochschild homology groups.
4. The Hochschild complex also has applications in deformation theory, where it helps understand how algebras can be deformed while preserving certain properties.
5. Connections between the Hochschild complex and cyclic homology are significant, as they provide insight into the relationship between algebraic structures and their geometric interpretations.

Review Questions

• How does the Hochschild complex relate to the concept of derived categories and what implications does this have for studying associative algebras?
  • The Hochschild complex can be viewed within the framework of derived categories as it allows for a richer understanding of the morphisms between chain complexes. By relating algebras to their derived categories, one can investigate homological properties and compute invariants through derived functors. This connection not only simplifies many computations but also reveals deeper structural insights into the behavior of associative algebras in various contexts.
• Discuss the significance of the spectral sequence arising from the Hochschild complex and its role in computing Hochschild homology groups.
  • The spectral sequence that originates from the Hochschild complex is essential for computing Hochschild homology groups as it organizes information about chain complexes into a manageable form. This powerful tool allows mathematicians to derive relationships between different cohomological groups step-by-step, facilitating calculations that might otherwise be too complex. The convergence of this spectral sequence demonstrates how intricate algebraic structures can be systematically analyzed through successive approximations.
• Evaluate the impact of the Hochschild complex on understanding deformation theory within algebraic structures and its broader implications.
  • The Hochschild complex significantly influences deformation theory by providing a framework for understanding how algebras can change while maintaining essential properties. Through its connections to cohomology, mathematicians can analyze deformations in terms of their higher-order obstructions. This understanding extends beyond mere algebraic manipulation, impacting areas like algebraic geometry and representation theory, where such deformations often yield insights into geometric and topological phenomena.

Related terms

Cohomology: A mathematical tool used to study topological spaces by associating algebraic objects, such as groups or rings, to these spaces, often revealing important structural properties.

Derived Category: A category that encodes information about chain complexes, allowing for the study of morphisms between complexes and facilitating computations in homological algebra.

Cyclic Homology: A variant of homology theory that is particularly useful in the context of algebras, capturing information about their structure through cyclic chains and cycles.