1
$\begingroup$

Given two finite dimensional algebra $A$ and $B$ with generator-cogenerators $M$ and $N$. Define two algebras $X:=End_A(M)$ and $Y:=End_B(N)$. Assume X and Y are derived equivalent. Are A and B derived equivalent?

It is an open problem whether $A$ is selfinjective iff $B$ is selfinjective, see conjecutre 3 of https://link.springer.com/article/10.1007%2Fs11856-016-1327-4.

A positive answers to my quesiton would prove this open problem to be true since derived equivalences preserve being selfinjective. (at least over an algebraically closed field)

edit: A generator is a module having every indecomposable projective module as a direct summand and dually a cogenerator is a module having any injective module as a direct summand. All modules are assumed to be finite dimensional.

$\endgroup$
10
  • $\begingroup$ Maybe I don't understand the terminology, but does ordinary Morita equivalence imply derived equivalence? If so then $X$ is derived equivalent to $A$ and similarly for $Y$ and $B$ so... yes? $\endgroup$ Oct 1, 2017 at 22:17
  • $\begingroup$ @DylanWilson I added the definitions of generator and cogenerator. Of course M will be no progenerator in general and thus A and X are not Morita equivalent. $\endgroup$
    – Mare
    Oct 1, 2017 at 22:29
  • 4
    $\begingroup$ Why would they be derived equivalent? You have a certain tendency to ask questions in what looks like a fishing expedition... $\endgroup$ Oct 2, 2017 at 4:33
  • 1
    $\begingroup$ Yes. And why would they be derived equivalent in general? (My point, Mare, in case you are not picking it is that all this information belongs in the question. If you have reasons to expect an affirmative answer, or for want it to be true, or whatever, explain them in the question. You did not even spell out the fact that an affirmative answer to your question would answer the open problem (which clearly puts your question in the realm of open questions, which turn to be, quite often, very bad questions for MO)) $\endgroup$ Oct 2, 2017 at 18:30
  • 1
    $\begingroup$ @MarianoSuárez-Álvarez I ask the question because there could be an easy counterexample. If it is true, then of course it would be hard to prove since it answers the open problem. $\endgroup$
    – Mare
    Oct 2, 2017 at 18:42

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.