A Lower-Bound on the Hochschild Cohomological Dimension

TL;DR

This paper establishes a new lower-bound for the Hochschild cohomological dimension of commutative algebras, revealing that many are not quasi-free even if smooth, thus extending previous results to broader base rings.

Contribution

It provides a concrete lower-bound for Hochschild cohomological dimension in terms of other invariants, generalizing prior results to arbitrary commutative rings with unity.

Findings

Most $k$-algebras are not quasi-free even if smooth

The lower-bound relates Hochschild cohomology to other homological invariants

Generalizes previous results from complex numbers to arbitrary rings

Abstract

A concrete lower-bound for the Hochschild cohomological dimension of a commutative $k$-algebra, in terms of three other homological invariants is obtained. This result is then used to show that most $k$-algebras fail to be quasi-free, even if they are smooth. This result generalizes a result of \cite{cuntz1995algebra} to the case where the base-ring is no longer $\cc$ but can be any commutative ring with unity.