Hochschild Cohomological Dimension is Not Upper Semi-Continuous

TL;DR

This paper demonstrates that Hochschild cohomological dimension is not upper semi-continuous in non-commutative algebras, providing a counterexample to the semi-continuity theorem in this context.

Contribution

It shows that the semi-continuity theorem fails for non-commutative algebras by constructing a specific family of algebras with varying Hochschild cohomological dimensions.

Findings

Most algebras in the family have Hochschild cohomological dimension 2

One algebra in the family has Hochschild cohomological dimension 1

The semi-continuity theorem does not hold for non-commutative algebras

Abstract

It is shown that the Hochschild Cohomological dimension of an associative algebra is not an upper-semi continuous function, showing the semi-continuity theorem is no longer valid for non-commutative algebras. A family of $\mathbb{C}$ exhibits this-algebras parameterized by $\mathbb{C}$ all but one of which has Hochschild cohomological dimension $2$ and the other having Hochschild cohomological dimension $1$.