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Search: MSC category 16G60 ( Representation type (finite, tame, wild, etc.) )

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1. CJM 2007 (vol 59 pp. 332)

Leuschke, Graham J.
Endomorphism Rings of Finite Global Dimension
For a commutative local ring $R$, consider (noncommutative) $R$-algebras $\Lambda$ of the form $\Lambda = \operatorname{End}_R(M)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $\Lambda$ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of $\operatorname{Spec} R$. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal $\mathbb{C}$-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra $\Lambda$ with finite global dimension and which is maximal Cohen--Macaulay over $R$ (a ``noncommutative crepant resolution of singularities''). We produce algebras $\Lambda=\operatorname{End}_R(M)$ having finite global dimension in two contexts: when $R$ is a reduced one-dimensional complete local ring, or when $R$ is a Cohen--Macaulay local ring of finite Cohen--Macaulay type. If in the latter case $R$ is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.

Keywords:representation dimension, noncommutative crepant resolution, maximal Cohen--Macaulay modules
Categories:16G50, 16G60, 16E99

2. CJM 2006 (vol 58 pp. 180)

Reiten, Idun; Ringel, Claus Michael
Infinite Dimensional Representations of Canonical Algebras
The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Pr\"{u}fer modules, and how other modules are determined by these modules.

Categories:16D70, 16D90, 16G20, 16G60, 16G70

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