# Hochschild cohomology of commutative quotients

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)?

• I suspect that you can get counterexamples with d=1 and A=k[t] with x=t^m for suitable m -- but could you please clarify what you mean by N/I here? – Yemon Choi Jan 2 '15 at 0:46
• By N/IN i mean the left submodule generated by I – AIM_BLB Jan 2 '15 at 0:53
• Intuitively it seems over-optimistic to ask for A/I to have smaller homological dimension than A. IIRC you can find explicit, periodic resolutions of $k[t]/t^m k[t]$ in the books of Weibel and Loday, and so you can test tour conjecture on these. – Yemon Choi Jan 2 '15 at 0:57

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), therefore it has Hochschild cohomological dimension at most 1; whence in particular: $HH^{4}(\mathbb{C}\left<x\right>,A/IA)\cong 0$.