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In the theory of finite dimensional algebras there are many homological conjectures. When working over an algebraically closed field it is well known that any such algebra is Morita equivalent to a quiver algbra $kQ/I$ and thus it is enough to prove such a homological conjecture for quiver algebras. However, there are examples where the conjecture is known for quiver algebras but not for general algebras ,see for example the strong no loop conjecture https://www.sciencedirect.com/science/article/pii/S0001870811002714 .

Question:

Is there a homological result (which might have been a conjecture at some point, but not necessarily) that is known to hold true for quiver algebras but is false for general finite dimensional algebras?

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  • $\begingroup$ By definition, the answer to your question about conjectures is NO. $\endgroup$
    – Fernando Muro
    Commented Sep 14, 2018 at 13:41
  • $\begingroup$ I changed the question a bit. $\endgroup$
    – Mare
    Commented Sep 14, 2018 at 17:45

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