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Let $A$ be the algebra $K[x,y]/(x^2,y^2,xy,yx)$. Then $A$ is a 3-dimensional commutative algebra and $Ext^i(M,A) \neq 0$ for any indecomposable non-projective module $M$ for some $i>0$. Namely: $Ext^2(M,A)=Ext^1(\Omega^1(M),A) \neq 0$, since $\Omega^1(M)$ is a direct sum of simple modules.

Now do we have the property

$Ext_B^i(M,B) \neq 0$ for any indecomposable non-projective module $M$ for some $i>0$ (*)

also for the enveloping algebra $B=A^{op} \otimes_K A$ ?

This is a 9-dimensional commutative algebra with Loewy length 3. The same trick does not work since $\Omega^1(M)$ is just a Loewy length at most 2 module. But maybe one can classify the $B$-modules that are first (or higher) syzygy modules and show (*)?

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    $\begingroup$ You have posted a hundred questions on problems from homological algebra. I wonder what the point of all these questions is ? Are these questions from your research ? If so, what do you expect from MO ? Do you want the community to do your research for you ? Or should all these questions together be seen as part of a joint project ? I'm really puzzled on your motivation for posting all these questions. $\endgroup$
    – Todd Leason
    Commented Sep 7, 2017 at 13:55
  • $\begingroup$ @ToddLeason Yes those are questions from my research. I have alot of questions and like to share them usually. But MO does not do my research. I just have one paper online at the moment where MO helped me with a result as far as I can remember. If someone can help me with a hard problem, he is always invited to join to write an article so you can see those questions as a joint project if you want. I think only around 10% of the question/conjectures I have are posted on MO. $\endgroup$
    – Mare
    Commented Sep 7, 2017 at 14:10

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