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Finite and countable CM-representation type. (English) Zbl 0719.14024

Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht/Pfalz/FRG 1985, Lect. Notes Math. 1273, 9-34 (1987).
[For the entire collection see Zbl 0619.00007.]
Theorem [cf. H. Knörrer, Invent. Math. 88, 153-164 (1987; Zbl 0617.14033) and R.-O. Buchweitz, G.-M. Greuel and the author, ibid. 165-182 (1987; Zbl 0617.14034)]: A hypersurface singularity is simple iff it has finite CM-representation type, i.e. iff there are only finitely many isomorphism classes of indecomposable maximal Cohen- Macaulay modules.
I include a complete proof of this result in this survey article. My emphasis lies on the construction of maximal Cohen-Macaulay modules via matrix factorizations (section 2), from which we deduce Knörrer’s periodicity result (section 3). In section 4 we construct for every nonsimple hypersurface singularity infinitely many indecomposable maximal Cohen-Macaulay modules.
Theorem [cf. the author’s joint paper with Buchweitz and Greuel cited above]: A hypersurface has countable CM-representation type iff it is isomorphic to \(A_{\infty}\) or \(D_{\infty}.\)
In section 6 we describe all indecomposable maximal Cohen-Macaulay modules on \(A_ n\) and \(D_ n\) singularities for \(n=1,2,...,\infty\) and their Auslander-Reiten-sequences. - A complete classification of complex analytic singularities with finite CM-representation type is known only for dimension \(\leq 2\). A list of the known examples is contained in (7.1). In (7.2) I give a few more examples of singularities which have countable representation type.

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14B05 Singularities in algebraic geometry