Localization in Hochschild homology

TL;DR

This paper explores a sheaf-theoretic approach to localization in Hochschild homology within noncommutative geometry, emphasizing real algebraic geometry techniques to analyze complex algebras through simpler components.

Contribution

It introduces a novel sheafification-based localization method in Hochschild homology using real algebraic geometry, enabling analysis of complex algebras via simpler stalks.

Findings

Sheafification of algebras simplifies Hochschild homology computations.

Reduction to stalks allows analysis of complex algebras from simpler cases.

Method bridges real algebraic geometry with noncommutative geometry techniques.

Abstract

Localization methods are ubiquitous in cyclic homology theory, but vary in detail and are used in different scenarios. In this paper we will elaborate on a common feature of localization methods in noncommutative geometry, namely sheafification of the algebra under consideration and reduction of the computation to the stalks of the sheaf. The novelty of our approach lies in the methods we use which mainly stem from real instead of complex algebraic geometry. We will then indicate how this method can be used to determine the Hochschild homology theory of more complicated algebras out of simpler ones.