On the growth of the Betti sequence of the canonical module

Joint work with David Jorgensen.

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We study the growth of the Betti sequence of the canonical module of a Cohen–Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein. We answer the question of exponential growth affirmatively for a large class of rings, and prove that the growth is in general not extremal. As an application of growth, we give criteria for a Cohen–Macaulay ring possessing a canonical module to be Gorenstein.

Several years ago, when I was a postdoc at Kansas, Dave Jorgensen spent a couple of summers visiting Lawrence to work with Craig Huneke and Dan Katz. Somewhere in there, we heard from Craig the main question this paper tries to answer: If the Betti numbers of the canonical module are bounded, must it have finite projective dimension? I believe it was motivated by some work that Craig had been doing with Doug Hanes. The question has stuck in our craws ever since: it’s so easy to state, but we’ve never been able to get a good clear answer to it. Many hundreds of examples lead us to believe it’s true. After three or four years of batting it back and forth intermittently, we felt like we had made enough progress for a short paper.

Three results appearing in Section 2 are incorrect as stated. The second part of Lemma 2.1, which assumes that $\Ext^i_R(M,N^\vee)=0$ for all $i$ in a certain range andconcludes an inequality on the Betti numbers of $N$, is not true for the stated range of $\Ext$-vanishing. The correction forces changes in the statements of Theorems 2.2 and 2.4 as well. Here is an erratum explaining and repairing the mistake.

The erratum above also fixes a bit of sloppy writing in the same Lemma 2.1, where the assumptions on n weren’t exactly clear. Thanks to Roger Wiegand for pointing this out to us.

Bibtex code:

 @article {MR2299575,
     AUTHOR = {Jorgensen, David A. and Leuschke, Graham J.},
      TITLE = {On the growth of the {B}etti sequence of the canonical module},
    JOURNAL = {Math. Z.},
   FJOURNAL = {Mathematische Zeitschrift},
     VOLUME = {256},
       YEAR = {2007},
     NUMBER = {3},
      PAGES = {647–659},
       ISSN = {0025-5874},
      CODEN = {MAZEAX},
    MRCLASS = {13Dxx},
   MRNUMBER = {MR2299575},