Local rings of bounded Cohen-Macaulay type

Joint work with Roger Wiegand.

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It is known that a one-dimensional CM local ring R has finite CM type if and only if R is reduced and has bounded CM type. Here we study the one-dimensional rings of bounded but infinite CM type. We classify these rings up to analytic isomorphism (under the additional hypothesis that the ring contains an infinite field). In the first section we deal with the complete case, and in the second we show that bounded CM type ascends to and descends from the completion. In the third section we study ascent and descent in higher dimensions and prove a Brauer-Thrall theorem for excellent rings.

This is the second of a pair of papers I wrote with Roger Wiegand, completely settling the question of which equicharacteristic one-dimensional rings have a bound on the ranks of indecomposable torsion-free modules. It’s a little surprising that the answer is almost completely contained in the first paper (Hypersurfaces of bounded Cohen-Macaulay type): in addition to the two hypersurfaces, there is only one additional isomorphism class. The second section, on ascent to and descent from the completion, is very satisfying.

Bibtex code:

 @article {MR2162283,
     AUTHOR = {Leuschke, Graham J. and Wiegand, Roger},
      TITLE = {Local rings of bounded {C}ohen-{M}acaulay type},
    JOURNAL = {Algebr. Represent. Theory},
   FJOURNAL = {Algebras and Representation Theory},
     VOLUME = {8},
       YEAR = {2005},
     NUMBER = {2},
      PAGES = {225–238},
       ISSN = {1386-923X},
      CODEN = {ARTHF4},
    MRCLASS = {13C14 (13H10 13H15)},
   MRNUMBER = {MR2162283 (2006c:13013)},
 MRREVIEWER = {Geoffrey D. Dietz},