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The classification of special Cohen–Macaulay modules

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Abstract

In this paper we completely classify all the special Cohen–Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras.

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Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

    Osamu Iyama & Michael Wemyss

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  1. Osamu Iyama
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  2. Michael Wemyss
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Correspondence to Osamu Iyama.

Additional information

M. Wemyss was supported by the Cecil King Travel Scholarship, and would like to thank both the London Mathematical Society and the Cecil King Foundation.

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Iyama, O., Wemyss, M. The classification of special Cohen–Macaulay modules. Math. Z. 265, 41–83 (2010). https://doi.org/10.1007/s00209-009-0501-3

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