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Research/Hypersurfaces of Bounded Cohen-Macaulay Type

Joint work with Roger Wiegand.

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Abstract:

We investigate which rings of the form R = k[[x_0,\dots,x_d]]/(f), where k is a field and f is a non-zero non-unit of the formal power series ring, have bounded Cohen-Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen-Macaulay modules. As with finite Cohen-Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen-Macaulay type if and only if R \cong k[[x_0,\dots,x_d]]/(g+x_2^2+\dots+x_d^2), where g \in k[[x_0,x_1]] and [[x_0,x_1]]/(g) has bounded Cohen-Macaulay type. Up to analytic isomorphism, only two of the form k[[x_0,x_1]]/(g) have bounded Cohen-Macaulay type: g=x_1^2 and g=x_0x_1^2.

Notes:

This is the first of a pair of papers that I wrote with Roger Wiegand in 2002. There were two particularly fun parts: the general construction that shows that every hypersurface of bounded Cohen-Macaulay type comes from one of dimension one, and the delicate calculations that rule out most candidates in dimension one. Roger and I passed this manuscript back and forth by email for months (changing it from LaTeX to AMSTeX and back again each time).

Don’t miss the “Note added in proof” just before the references: The normal form needs a little adjustment when the residue field is not algebraically closed. Also, there is an error in the proof of the classification in dimensions higher than one. While the statement of the main theorem is correct, the argument in the paper does not cover the rings k[[x_0, ..., x_d]]/(x_d^2), where d > 1. Indeed, these rings do not have bounded CM type. Here is a proof of a more general result.

Bibtex code: 

 @article {MR2158755,
     AUTHOR = {Leuschke, Graham J. and Wiegand, Roger},
      TITLE = {Hypersurfaces of bounded {C}ohen-{M}acaulay type},
    JOURNAL = {J. Pure Appl. Algebra},
   FJOURNAL = {Journal of Pure and Applied Algebra},
     VOLUME = {201},
       YEAR = {2005},
     NUMBER = {1–3},
      PAGES = {204—217},
       ISSN = {0022–4049},
      CODEN = {JPAAA2},
    MRCLASS = {13C14 (13C05 13H10 13H15)},
   MRNUMBER = {MR2158755 (2006c:13014)},
 MRREVIEWER = {Geoffrey D. Dietz},
 }

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