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The Tachikawa conjecture states that $Ext^i(M,M) \neq 0$ for some $i \geq 1$ for every non-projective module $M$ over a selfinjective finite dimensional algebra. In theorem 4.6. of http://maths.nju.edu.cn/~huangzy/When%20are%20torsionless%20modules%20projective.pdf , the authors prove something (much stronger!) which implies the Tachikawa conjecture in the commutative case, which would be a sensational result in my opinion.

Question: Is the proof really true/without gaps? I couldnt understand everything and the authors never replied. Of course it might be extremely hard to give counterexamples to theorem 4.6. there but maybe gaps could be pointed out?

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On page 2163, the second line, the authors say "therefore ... is exact". I, and some others, believe this is a gap. The authors have been contacted (in 2010) and have not clarified.

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  • $\begingroup$ Interesting, did you make it public or what to do now? $\endgroup$
    – Mare
    Feb 11, 2017 at 21:07
  • $\begingroup$ I think one cannot really say that this spot is wrong in the selfinjective case since not even a nonprojective module M with $Ext^{1}(M,M)=0$ is known. $\endgroup$
    – Mare
    Feb 11, 2017 at 21:17
  • $\begingroup$ I didn't say it was wrong. I said I believe it is a gap, that is, not fully justified. $\endgroup$
    – Graham Leuschke
    Feb 13, 2017 at 16:20

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