# Selforthogonal modules over Artinian Gorenstein Rings.

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?

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

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".