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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 $K<x,y>/(x^2,y^2,xy-qyx)$ 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.

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