Wild Hypersurfaces

Joint work with Andrew Crabbe.

Download Paper (revised version of 2011-03-08)

Complete hypersurfaces of dimension at least 2 and multiplicity at least 4 have wild Cohen-Macaulay type.

Andrew was a postdoc here at Syracuse 2008–2010. We worked on a couple of projects, but this was the one we spent the most time on. It began as an effort to understand Bondarenko’s result that two-dimensional hypersurfaces of multiplicity four or more have wild representation type. In working through his proof, we found some simplifications that allowed the argument to actually work for all dimensions at least two. For the higher dimensions, the result seems to be “morally” known thanks to earlier work of Buchweitz-Greuel-Schreyer, but we thought it was worth writing down explicitly. Among other things, it lets us introduce the idea of tameness and wildness to a commutative-algebraist audience, which I’ve wanted to do for a while.

Bibtex code:

@article {MR2811571,
    AUTHOR = {Crabbe, Andrew and Leuschke, Graham J.},
     TITLE = {Wild hypersurfaces},
   JOURNAL = {J. Pure Appl. Algebra},
  FJOURNAL = {Journal of Pure and Applied Algebra},
    VOLUME = {215},
      YEAR = {2011},
    NUMBER = {12},
     PAGES = {2884--2891},
      ISSN = {0022-4049},
     CODEN = {JPAAA2},
   MRCLASS = {13C14 (13H10 16G60)},
  MRNUMBER = {2811571 (2012e:13018)},
MRREVIEWER = {Geoffrey D. Dietz},
       DOI = {10.1016/j.jpaa.2011.04.009},
       URL = {http://dx.doi.org/10.1016/j.jpaa.2011.04.009},