The F-signature and strong F-regularity

Joint work with Ian Aberbach.


We show that the F-signature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly F-regular.

Craig Huneke and I defined something in Two theorems about maximal Cohen-Macaulay modules called the F-signature of a local ring of characteristic p (originally we called it the “rational signature” – Craig wanted to get the word ‘signature’ in there somewhere – but it’s just as well we didn’t use that, since the concept turns out to have nothing to do with F-rationality). It’s caught on, which is immensely satisfying. Before the paper even appeared, Ian Aberbach emailed me to say he had some ideas about how to improve one of the theorems about the F-signature. He invited me to Missouri to give a talk and to chat about it. We worked out the details in a very swanky coffeehouse in Columbia.

Bibtex code:

 @article {MR1960123,
    AUTHOR = {Aberbach, Ian M. and Leuschke, Graham J.},
     TITLE = {The {$F$}-signature and strong {$F$}-regularity},
   JOURNAL = {Math. Res. Lett.},
  FJOURNAL = {Mathematical Research Letters},
    VOLUME = {10},
      YEAR = {2003},
    NUMBER = {1},
     PAGES = {51–56},
      ISSN = {1073-2780},
   MRCLASS = {13A35},
  MRNUMBER = {MR1960123 (2004b:13003)},
 MRREVIEWER = {Karen E. Smith},