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Maximal Cohen-Macaulay modules over surface singularities. (English) Zbl 1200.14011
Skowroński, Andrzej (ed.), Trends in representation theory of algebras and related topics. Proceedings of the 12th international conference on representations of algebras and workshop (ICRA XII), Toruń, Poland, August 15–24, 2007. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-062-3/hbk). EMS Series of Congress Reports, 101-166 (2008).
The paper contains a lot of material and information about maximal Cohen–Macaulay modules \(M\) over the local ring \(A\) of a surface singularity (\(\dim (A)=\text{depth}(M)\)). It starts with basic results and definitions as for instance the depth lemma and the Auslander–Buchsbaum formula, contains Matlis Duality, Grothendieck’s Local Duality and a lot of other stuff from commutative algebra related to the study of maximal Cohen–Macaulay modules. Then general properties of maximal Cohen–Macaulay modules over surface singularities are presented as for instance the fact that in case \(A\) is normal \(M\) is maximal Cohen–Macaulay if and only if it is reflexive. It follows a section about maximal Cohen–Macaulay modules over two–dimensional quotient singularities containing the result that a normal surface singularity is a quotient singularity if and only if it has finite Cohen–Macaulay representation type.
The algebraic and the geometric approaches to McKay correspondence for quotient surface singularities as well as its generalization for simply elliptic and cusp singularities are described. A new proof of a result of R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer [Invent. Math. 88, 165–182 (1987; Zbl 0617.14034)] (the surface singularities \(A _{\infty}\) and \(D_{\infty}\) have countable Cohen–Macaulay representation type) is given. At the end one can find some conjectures concerning the Cohen–Macaulay representation type and an example for a Singular–computation.
For the entire collection see [Zbl 1144.16003].

14B05 Singularities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C14 Cohen-Macaulay modules
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