Hochschild Cohomological Dimension is Not Upper Semi-Continuous
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theorem is no longer valid for non-commutative algebras.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 $1$.