Cohen-Macaulay Representations

in press, to appear in AMS series Mathematical Surveys and Monographs

Roger and I wrote a book!

Joint work with Roger Wiegand.

We’ve posted all the errata we know about on the AMS web site. You are cordially invited to let us know of other goofs we’ve missed. Thank you!

This book is a comprehensive treatment of the representation theory of maximal Cohen-Macaulay (MCM) modules over local rings. This topic is at the intersection of commutative algebra, singularity theory, and representations of groups and algebras.

Two introductory chapters treat the Krull-Remak-Schmidt Theorem on uniqueness of direct-sum decompositions and its failure for modules over local rings. Chapters 3-10 study the central problem of classifying the rings with only finitely many indecomposable MCM modules up to isomorphism, i.e., rings of finite CM type. The fundamental material – ADE/simple singularities, the double branched cover, Auslander-Reiten theory, and the Brauer-Thrall conjectures – is covered clearly and completely. Much of the content has never before appeared in book form. Examples include the representation theory of Artinian pairs and Burban-Drozd’s related construction in dimension two; an introduction to the McKay correspondence from the point of view of maximal Cohen-Macaulay modules; Auslander-Buchweitz’s MCM approximation theory; and a careful treatment of non-zero characteristic. The remaining chapters 11–17 present results on bounded and countable CM type and on the representation theory of totally reflexive modules.

Kid-tested, kid-approved!

Coming soon.

Bibtex code:

@book {MR2919145,
    AUTHOR = {Leuschke, Graham J. and Wiegand, Roger},
     TITLE = {Cohen-{M}acaulay representations},
    SERIES = {Mathematical Surveys and Monographs},
    VOLUME = {181},
 PUBLISHER = {American Mathematical Society},
   ADDRESS = {Providence, RI},
      YEAR = {2012},
     PAGES = {xviii+367},
      ISBN = {978-0-8218-7581-0},
   MRCLASS = {13C14 (16Gxx)},
  MRNUMBER = {2919145},

Here are some links: