Connes' Exact Sequence

Definition

Connes' Exact Sequence is a powerful tool in the study of Hochschild and cyclic homology, providing a way to relate these two important concepts in algebraic topology and noncommutative geometry. This sequence captures the relationship between the Hochschild homology of algebras and their cyclic homology, revealing deeper connections between these areas. It is essential for understanding how cyclic homology can be viewed as a refinement of Hochschild homology, bridging gaps between algebraic and topological properties.

5 Must Know Facts For Your Next Test

1. Connes' Exact Sequence relates the Hochschild homology of an algebra to its cyclic homology by providing a long exact sequence that reveals their interactions.
2. The sequence typically takes the form: $$0 \to HH_n(A) \to HC_n(A) \to HC_{n-1}(A) \to HH_{n-1}(A) \to 0$$ for a given algebra A.
3. This exact sequence allows mathematicians to compute cyclic homology using known results from Hochschild homology, making it easier to derive important properties.
4. Connes' Exact Sequence plays a key role in noncommutative geometry, linking geometric objects to algebraic structures through homological methods.
5. The sequence demonstrates how cyclic homology can provide finer invariants than Hochschild homology, leading to richer interpretations in the study of algebras.

Review Questions

• How does Connes' Exact Sequence illustrate the relationship between Hochschild and cyclic homology?
  • Connes' Exact Sequence serves as a bridge between Hochschild and cyclic homology by providing an exact sequence that connects these two theories. It shows how Hochschild homology can be embedded within cyclic homology, revealing that the latter can be thought of as a more refined version of the former. This connection highlights how changes in algebraic structures can impact both types of homology, allowing for deeper insights into their respective properties.
• Discuss the implications of Connes' Exact Sequence in the field of noncommutative geometry.
  • In noncommutative geometry, Connes' Exact Sequence is significant because it links geometric concepts with algebraic structures through the lens of homological algebra. This connection allows researchers to apply techniques from topology and algebra to study spaces that cannot be described classically. The insights gained from this exact sequence can lead to a better understanding of how noncommutative spaces behave and can reveal new invariants that enrich our understanding of geometry.
• Evaluate the impact of Connes' Exact Sequence on computational techniques within algebraic topology.
  • Connes' Exact Sequence has transformed computational methods in algebraic topology by enabling mathematicians to calculate cyclic homology more effectively through known Hochschild results. This impact is profound as it streamlines computations and helps uncover relationships between various algebraic objects. Additionally, by providing a framework for comparing different types of homologies, it fosters deeper explorations into their respective invariants, ultimately broadening our comprehension of complex topological structures.

Related terms

Hochschild Homology: A homology theory for algebras that measures the 'size' of the space of multilinear maps and encodes information about the algebra's structure.

Cyclic Homology: A homology theory that extends Hochschild homology by taking into account cyclic permutations, focusing on the algebraic structure of noncommutative spaces.

Exact Sequence: A sequence of algebraic objects and morphisms between them that captures the idea of continuity and structure preservation, crucial for understanding relationships in homological algebra.