Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $R$ be a local artinian Gorenstein ring and $M$ a finitely generated $R$-module, then $Ext_R^1(M,M) = 0$ if only if $M$ is projective?

share|improve this question
    
Dear TmobiusX, would you care to explain any motivation for your question? –  Hailong Dao May 26 '10 at 16:17

1 Answer 1

up vote 2 down vote accepted

This would be a very strong version of the Auslander-Reiten Conjecture (see here, for example) in the Gorenstein case. The Conjecture is still open, though many partial results are known.

By the way, this paper (ScienceDirect link, may not be visible to everyone) claims a proof of the Conjecture in the Gorenstein case, which would give an affirmative answer to your question, but I --- and several other people I've talked to --- believe there is a gap in the proof. Their assumption is that $\mathrm{Ext}_R^{i}(M,M)=0$ for $i =1,2$. The questionable step is at the top of page 2163, the second line, where they say "therefore ... is exact".

share|improve this answer
    
Thank you, Graham. –  TmobiusX May 27 '10 at 1:07

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.