**Notation:**
Let $k$ be a commutative local ring and let $HH^{i}(A,N)$ denote the $i^{th}$ Hochschild cohomology $k$-module of a $k$-algebra A with coefficients in an $(A,A)$-bi-module $N$.

If $x:=\{x_1,...,x_d\}$ is a maximal regular sequence in $A$ then is it true that:
$HH^{i+d}(A,N)$ vanishes *only if* $HH^{i}(A/I,N/IN)$ vanishes?

If so how can this be proven? (if not was is the prelationship between the hochschild cohomological dimebsion of A and A/I)?