Reduction of isolated singularities
type: Item
instance of: scholarly article
title: Reduction of isolated singularities
zbMATH Open document ID: 0654.14002
DOI: 10.1007/BF02564469
author: Ernst Dieterich
published in: Commentarii Mathematici Helvetici
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140105
review text: Let \(f\in {\mathbb{C}}[X_ 0,...,X_ d]\) be a polynomial such that the hypersurface defined by f has an isolated singularity. Denote by A the complete local ring \({\mathbb{C}}X_ 0,...,X_ d/(f)\). It is known that A has a simple singularity iff A has only finitely many isomorphism classes of indecomposable Cohen-Macaulay modules (Knörrer, Buchweitz- Greuel-Schreyer). In this case the Auslander-Reiten quiver Q(A) of A is determined and turns out to be closely related to the Dynkin diagram of A. The author begins the study of the category of Cohen-Macaulay A-modules in the case when A has an isolated but not a simple singularity, proving the following results: (1) Let C be the connected component of Q(A) containing A. Then \(Q(A)- C={\dot \cup}_{i\in I}A_{\infty}/<x^{n(i)}>,\) where I is an index set and \(n(i)\in\{1,2\}\). Moreover, if d is even, \(n(i)=1\) for all i. (2) There exists an arithmetic sequence of positive integers r, 2r, 3r,... such that for each mr \((m\geq 1)\) there exists an infinite sequence \((M_{m,n})_{n\geq 1}\) of indecomposable pairwise non-isomorphic Cohen- Macaulay A-modules all of which of rank mr.
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 4067075
zbMATH Keywords: simple singularity
zbMATH Keywords: Cohen-Macaulay modules
zbMATH Keywords: Auslander-Reiten quiver
zbMATH Keywords: Dynkin diagram
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
OpenAlex ID: W2081871800
description: scientific article; zbMATH DE number 4067075