Cohen-Macaulay representations
type: Item
instance of: scholarly article
zbMATH Open document ID: 1252.13001
author: Graham J. Leuschke
author: Roger Wiegand
publication date: 2012-05-30T00:00:00Z
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 13C14
zbMATH DE Number: 6039453
zbMATH Keywords: maximal Cohen-Macaulay module
zbMATH Keywords: finite Cohen-Macaulay representation type
zbMATH Keywords: isolated singularity
zbMATH Keywords: hypersurface ring
zbMATH Keywords: Auslander-Reiten sequences
zbMATH Keywords: Auslander-Reiten quiver
zbMATH Keywords: MCM approximations
MaRDI profile type: Publication
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title: Cohen-Macaulay representations
published in: Mathematical Surveys and Monographs
review text: The book under review is mainly concerned with the \textit{problem} of the classification of Cohen-Macaulay local rings \((R,\, \mathfrak{m},\, k)\) having only a finite number of indecomposable maximal Cohen-Macaulay module (one says that such a ring has \textit{finite CM type}). Compared with the canonical reference for the subject, namely the book of \textit{Y. Yoshino} [Cohen-Macaulay modules over Cohen-Macaulay rings. London Mathematical Society Lecture Note Series. 146. Cambridge (UK): Cambridge University Press (1990; Zbl 0745.13003)], it contains a number of new results and topics. Among them, the elementary proof of \textit{C. Huneke} and \textit{G. J. Leuschke} [Math. Ann. 324, No. 2, 391--404 (2002; Zbl 1007.13005)] of the theorem of \textit{M. Auslander} [Lect. Notes Math. 1178, 194--242 (1986; Zbl 0633.13007)] asserting that if \(R\) has finite CM type then \(R_{\mathfrak{p}}\) is a regular local ring, \(\forall \, \mathfrak{p} \in \text{Spec}\, R \setminus \{\mathfrak{m}\}\), and a proof, due to \textit{R. Wiegand} [J. Algebra 203, No. 1, 156--168 (1998; Zbl 0921.13015)] and \textit{G. Leuschke} and \textit{R. Wiegand} [J. Algebra 228, No. 2, 674--681 (2000; Zbl 0963.13020)] of a conjecture of \textit{F.-O. Schreyer} [Lect. Notes Math. 1273, 9--34 (1987; Zbl 0719.14024)] asserting that \(R\) has finite CM type if and only if its \(\mathfrak{m}\)-adic completion \(\widehat R\) has finite CM type. Actually, for the proof of the ``only if'' part of the later result, the authors have to assume that \(R\) is excellent (it is unknown whether this condition is necessary or not). Besides, the proof of the ``only if'' part depends on a difficult result of \textit{R. Elkik} [Ann. Sci. Éc. Norm. Supér. (4) 6, 553--603 (1973; Zbl 0327.14001)] for which the authors refer to Elkik's paper.NEWLINENEWLINEFor the majority of the theorems contained in the book under review the authors provide complete and (essentially) self-contained proofs, preferring explicit construc…
reviewed by: Iustin Coandă
description: scientific article; zbMATH DE number 6039453