This question concerns the following Lemma 4.2 in this paper by Van den Bergh:

Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\text{End}_R(M)$ is maximal Cohen-Macaulay (i.e. $A$ has maximum depth, in this situation $A$ is often called an $R$-order). Suppose $\text{gl.dim} (A)<\infty$, then $\text{gl.dim}(A)=d$.

My question is, does the Gorenstein hypothesis necessary for dimension $d>2$:

Is the above Lemma true with $R$ being normal and Cohen-Macaulay, assuming $d>2$?


If not, is there a upper bound on $\text{gl.dim} (A)$? ($d$ is an obvious lower bound)

(The proof in quoted paper used Gorensteiness in the second sentence: "$\text{Ext}^i_R(A,R)=0$ for $i>0$..." Under mild hypotheses one can try to replace $R$ by the canonical module, but then run into troubles in line 3...).

  • $\begingroup$ I've thought about this a bit too, and never found anything good to say. Another obvious way out of the pickle is to ask for $A$ to be totally reflexive as an $R$-module. If I recall correctly, this formally gives the same good results (the center has rational singularities, e.g.) but it's hard for me to see whether any interesting examples exist, never mind whether it can be done in any generality. $\endgroup$ – Graham Leuschke Jul 25 '10 at 11:24
  • $\begingroup$ @Graham: thanks, I will think more about that. May be I will just email Van den Bergh. $\endgroup$ – Hailong Dao Jul 26 '10 at 14:29

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