leuschke.org :: Research/On a Conjecture of Auslander and Reiten

Research/On a Conjecture of Auslander and Reiten

Joint work with Craig Huneke.

Abstract:

We prove a conjecture of Auslander and Reiten for a large class of Cohen-Macaulay rings. The conjecture, proposed in 1975, states that if M is a finitely generated R-module such that Extⁱ_R(M \oplus R , M \oplus R)=0 for all i > 0 then M is projective. We establish the conjecture for excellent Cohen-Macaulay normal domains containing the rational numbers, and slightly more generally.

Erratum:

There is a small mistake in the proof of the main theorem, which necessitates a small change in the statement. At the beginning of the proof of Theorem 1.3, we say that the case d=1 follows from results of Auslander-Ding-Solberg. To apply the ADS results, however, we need one additional Ext to vanish: we have to assume both Ext¹(M,M) and Ext²(M,M) are zero. Thus the statement of Theorem 1.3 should include this assumption in the case d=1. Of course, in the context of the Auslander-Reiten conjecture, this is not a major problem.

Notes:

The first version of this paper was an utter failure. I had an argument using spectral sequences (which I learned about from Foxby’s paper “On the μ_i…”) that I thought gave the full result. I spoke on the theorem several places (including Illinois, when I visited there around Thanksgiving 2001), but always swept the details under the rug (because who wants to see a spectral-sequence argument?). Finally Lucho Avramov sat me down and asked to see the details, and immediately pointed out where I had gone wrong. I was mortified. Luckily, Craig got interested in the question, and convinced me to keep thinking about it. (In fact, I’d originally heard the question from him, and he’d learned it from Ragnar-Olaf Buchweitz.) We were able to put together a very nice argument to answer almost all cases. The assumptions that the ring be a quotient of a normal domain by a regular sequence, and/or contain a field, are still quite frustrating, though, since they obviously have nothing to do with the question.

Bibtex code:

 @article {MR2052636,
    AUTHOR = {Huneke, Craig and Leuschke, Graham J.},
     TITLE = {On a conjecture of Āuslander and {R}eiten},
   JOURNAL = {J. Algebra},
  FJOURNAL = {Journal of Algebra},
    VOLUME = {275},
      YEAR = {2004},
    NUMBER = {2},
     PAGES = {781—790},
      ISSN = {0021–8693},
     CODEN = {JALGA4},
   MRCLASS = {13D07 (13H10)},
  MRNUMBER = {MR2052636 (2005a:13033)},
 MRREVIEWER = {Juan Ramon Garc{\’{\i}}a Rozas},
 }

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