# Question on vanishing Hochschild cohomology

Recall that for an $$K$$-algebra $$A$$ with $$A^e:=A^{op} \otimes_K A$$ the Hochschild cohomology is defined as $$HH^n(A,M):=Ext_{A^e}^n(A,M)$$.

Question:

Is there a finite dimensional selfinjective algebra $$A$$ with $$HH^n(A,A)=0$$ for all $$n \geq t$$ for some $$t$$ and such that there are at most finitely many indecomposable $$A$$-bimodules with $$HH^i(A,M)=0=HH^1(A,\Omega^i(M))$$ for all $$i \geq 1$$ (or even better no non-projective indecomposable $$A$$-modules $$M$$ with this property)?

An example of such $$A$$ is $$K/(x^2,y^2,xy-qyx)$$ for a non-root of unity $$q$$, see http://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2005/0012/0006/MRL-2005-0012-0006-a002.pdf.

It is hard to do experiments with the computer, but it seems to me that the condition $$0=HH^1(A,\Omega^i(M))$$ for all $$i \geq 1$$ is rather strong. For example $$HH^1(A,\Omega^1(M))=\underline{Hom_{A^e}}(A,M)$$ should often be non-zero.

Note that this is a special case of question 2 of Gorenstein projective modules of a certain triangular matrix algebra , where a positive answer would have an interesting application as exlained in this thread.