1. Introduction and notation

Let $(R, \underline{m})$ be either a complete local Noetherian commutative algebra over a field K or a graded K-algebra $R=\oplus_{d\leq 0} R_d$, R₀=K and $\underline{m} =\oplus_{d>0} R_d.$ Let $M \in \mbox{Mod} \, R$ 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 $\mbox{End}(M)$ 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 $X=Proj\ R$, let V be a vector bundle on X with $M \in \mbox{Mod} \, R$ an associated graded R-module. The algorithm produces the direct sum decomposition of V.
• (i₂) More general, let $\cal{M}$ be a coherent sheaf on X. We can decide wether $\cal{M}$ is indecomposable.
• (i₃) Let R be the local ring of an isolated hypersurface singularity X defined by a polynomial equation $f(x_1,\ldots,x_n)$. If X is a suspension over X₀, i.e. $f=f_0(x_1,\ldots,x_{n-1})+x_n^t$ 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 $M\in MCM(X)$ and R_i:=R/x_nⁱ⁺¹R, $M_i:=M \otimes R_i$ then $syz_R(M_{t-2}) \cong M \oplus syz_R(M)$ and M_t-2 is a deformation of M₀ over R_t-2.
  
  Knowing $M_0\in MCM(X_0)$ 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 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.

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 $Q \in \mbox{Mat}(r,n;R)$