Maximal orders of global dimension and Krull dimension two
type: Item
instance of: scholarly article
title: Maximal orders of global dimension and Krull dimension two
zbMATH Open document ID: 0591.16002
DOI: 10.1007/BF01388739
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143340
review text: Let R be a (commutative) complete Noetherian two-dimensional local ring, with an algebraically closed residue field k of characteristic zero. The object of this paper is to classify the algebras A which are maximal R- orders and have global dimension 2. The case where \(A=R\) (so R is normal) is known - R is then a quotient singularity, so \(R=ks,t^ G\) where G is a small subgroup of \(GL_ 2(k)\), [\textit{J. Herzog}, Math. Ann. 233, 21-34 (1978; Zbl 0358.13009), \textit{H. Esnault}, J. Reine Angew. Math. 362, 63-71 (1985; Zbl 0553.14016), \textit{M. Auslander}, Almost split sequences and rational singularities (preprint)]. By a theorem of M. Auslander, this classification problem is equivalent to classifying the maximal R-orders in a division ring which have only finitely many isomorphism classes of indecomposable Cohen- Macaulay modules. The author classifies these orders, first for the case where R is a power series ring, and finally for the general case where R is a quotient singularity.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 3948515
zbMATH Keywords: maximal R-orders
zbMATH Keywords: global dimension
zbMATH Keywords: quotient singularity
zbMATH Keywords: indecomposable Cohen-Macaulay modules
zbMATH Keywords: power series ring
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
cites work: Some Elementary Examples of Unirational Varieties Which are Not Rational
cites work: Reflexive modules over rational double points
cites work: Rational Singularities and Almost Split Sequences
cites work: Maximal Orders
cites work: Rationale Singularitäten komplexer Flächen
cites work: Homologically Homogeneous Rings
cites work: Structure of hereditary orders
cites work: Q3253968
cites work: Q5340151
cites work: Reflexive modules on quotient surface singularities.
cites work: Q5335830
cites work: Construction géométrique de la correspondance de McKay
cites work: Rings mit only finitely many isomorphism classes of maximal indecomposable Cohen-Macaulay modules
cites work: Endliche Gruppen I
cites work: Tensor Products of Orders
cites work: Q3897070
cites work: Maximal Orders over Regular Local Rings of Dimension Two
cites work: Q4061092
cites work: Non-commutative affine P. I. rings are catenary
cites work: Théoremes de finitude en cohomologie galoisienne
cites work: Finite Unitary Reflection Groups
cites work: Q5181788
cites work: ON THE PURITY OF THE BRANCH LOCUS OF ALGEBRAIC FUNCTIONS
OpenAlex ID: W2017881838
author: Michael Artin
Recommended article: Correction to: Maximal orders of global dimension and Krull dimension two
Recommended article: Maximal orders over Krull domains
Recommended article: On the Kodaira dimension of maximal orders
Recommended article: Maximality of orders in Dedekind domains. II
Recommended article: Maximality of orders in Dedekind domains
Recommended article: On the global structure of regular orders of dimension two
Recommended article: Krull dimension and monomial orders
Recommended article: Q4012115
Recommended article: Krull dimension and global dimension of simple Ore-extensions
Recommended article: Krull and Global Dimensions of Fully Bounded Noetherian Rings
description: scientific article; zbMATH DE number 3948515