**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 $N$. 




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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)?



Hochschild cohomology of commutative quotients
No, it is false for example:


Let $A:=\mathbb{C}[x]/(x^4)$ and $N:=A$ then:
$HH^{4}(A,N)\cong \mathbb{C}[x]$ (Weibel: p.304)


Contrastingly: $\mathbb{C}[x]$ ($\cong \mathbb{C}\left<x\right>$) is quasi-free (see Cuntz-Quillen's [article](http://www.ams.org/journals/jams/1995-08-02/S0894-0347-1995-1303029-0/S0894-0347-1995-1303029-0.pdf)), therefore it has Hochschild cohomological dimension at most 1; whence in particular: $HH^{4}(\mathbb{C}\left<x\right>,A/IA)\cong 0$.


