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# 1. Introduction and notation

Let be either a complete local Noetherian commutative algebra over a field K or a graded K-algebra , R0=K and Let be a finitely generated (or graded resp.) R-module. In the local case it is known (cf. [Yo]) that the module M is indecomposable if and only if the endomorphism ring is local. But this criteria is difficult to handle, since it involves usage of non-commutative methods. Here we present several versions of a criteria based on a presentation matrix of M, which may be checked using only the Groebner-basis algorithm in various forms. The algorithm has several applications:
• (i1) 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.
• (i2) More general, let be a coherent sheaf on X. We can decide wether is indecomposable.
• (i3) Let R be the local ring of an isolated hypersurface singularity X defined by a polynomial equation . If X is a suspension over X0, i.e. and f0 an equation of X0, there is a useful theorem (cf. [HP] (2.8)) connecting maximal Cohen-Macaulay-modules (MCM) over X and X0 and generalizing Knörrer's periodicity theorem.

THEOREM If and Ri:=R/xni+1R, then and Mt-2 is a deformation of M0 over Rt-2.

Knowing we obtain MCM's over R by computing infinitesimal deformations of M0 and then inspecting the direct summands of their R-syzygy-module. In this way computations are partially done for X0 an Ak-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.
Throughout this paper we restrict the notation to the local case. The graded case is handled analogously.

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

Next: 2. Block type Bruhat Up: Splitting algorithm for vector Previous: Splitting algorithm for vector
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