The Classification of Special Cohen-Macaulay Modules

Understand

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.