Factoring the adjoint and MCM modules over the generic determinant

Joint work with Ragnar-Olaf Buchweitz.


G. M. Bergman recently asked whether the adjoint of the generic square matrix can be factored as a product of square matrices. He showed that in most cases there are no nontrivial factorizations. We recast the existence of such factorizations in terms of extensions of maximal Cohen–Macaulay modules over the hypersurface ring defined by the generic determinant. Specifically, a nontrivial factorization wherein one of the factors has determinant equal to the generic determinant gives an extension of the rank-one maximal Cohen–Macaulay (MCM) modules; we construct and classify all of these, as well as the corresponding factorizations of the adjoint. The classification shows that even in rank two, the MCM-representation theory of the generic determinant is quite wild. We also describe completely the Ext-algebra of the rank-one maximal Cohen–Macaulay modules.

Of all my research, this paper is the stuff that it’s most fun to give talks about. I gave colloquia and seminar talks about factoring the adjoint for nearly 4 years. I imagine people were starting to get tired of it, but it was just so much fun. The fact that it brought in “combing the hairy ball”, elementary matrix stuff that undergrads can follow, and concrete, explicit constructions with abstract-seeming things like Ext and free resolutions, made it a real pleasure to tell people about.

Bibtex code:

 @article {MR2343380,
    AUTHOR = {Buchweitz, Ragnar-Olaf and Leuschke, Graham J.},
     TITLE = {Factoring the adjoint and maximal {C}ohen-{M}acaulay modules
              over the generic determinant},
   JOURNAL = {Amer. J. Math.},
  FJOURNAL = {American Journal of Mathematics},
    VOLUME = {129},
      YEAR = {2007},
    NUMBER = {4},
     PAGES = {943–981},
      ISSN = {0002-9327},
     CODEN = {AJMAAN},
   MRCLASS = {13C14},
  MRNUMBER = {MR2343380},