Suppose A = C[[x,y,z]]/f(x,y,z) is one of the ADE singularities, where there are finitely many indecomposables P_1,...,P_n. In analogy with character theory of finite groups, we want to set up a situation where Hom(P_i,P_j) = delta _ij. That will allow us to decompose a reflexive A-module into a direct sum of indecomposables (in the same way one decomposes a representation into a direct sum of irreducible representations).
The triangulated category underline{CM}(A) = CM(A)/A has a t-structure with heart CM(A), in which the finitely many indecomposables are the simple objects. The simples satisfy Hom^0(S_i, S_j) = delta ij. To compute Hom^0 in this category use the equation underline{Hom}(M,N) simeq Ext^2_A(M,N). (See Burban and Drozd's survey paper, especially page 46.)
So basically you just have to compute Ext^2 in the complete local ring A. Singular can do this directly.
If you don't want to download Singular, Macaulay2 can do it, although it takes some care, because Macaulay naturally works with graded modules over polynomial rings; (one has to be careful with grading shifts.) For more information on the graded case, see the papers by [Kajiura Saito and Takahashi].
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