1. Introduction and notation

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*i*_{1}) Let*R*be the graded coordinate ring of a projective*K*-variety , let*V*be a vector bundle on*X*with an associated graded*R*-module. The algorithm produces the direct sum decomposition of*V*. - (
*i*_{2}) More general, let be a coherent sheaf on*X*. We can decide wether is indecomposable. - (
*i*_{3}) Let*R*be the local ring of an isolated hypersurface singularity*X*defined by a polynomial equation . If*X*is a suspension over*X*_{0}, i.e. and*f*_{0}an equation of*X*_{0}, there is a useful theorem (cf. [HP] (2.8)) connecting maximal Cohen-Macaulay-modules (MCM) over*X*and*X*_{0}and generalizing Knörrer's periodicity theorem.

THEOREM*If**and**R*_{i}:=*R*/*x*_{n}^{i+1}*R**,**then**and**M*_{t-2}*is a deformation of**M*_{0}*over**R*_{t-2}*.*

Knowing we obtain MCM's over R by computing infinitesimal deformations of*M*_{0}and then inspecting the direct summands of their*R*-syzygy-module. In this way computations are partially done for*X*_{0}an*A*_{k}-singularity (cf. [MPP]), which has stimulated the development of the algorithm. Our algorithm is used for the verification of the indecomposability of the module constructed there.

First we will fix some notation. A module *M* may be presented by matrices