# Decomposition of modules using computer packages

I am interested in computing direct sum decomposition of modules over some quotients of polynomial rings over a field (do not care much about the field at this point). Does any one know a package which does that? I tried searching Macaulay 2 but could not find any. Thanks.

EDIT: Perhaps the question was a bit vague, so I will add a specific example. Let $R=k[X_{11},...X_{33}]/I$, with $k$ say, $\mathbb Q$ or $\mathbb Z/(p)$ with $p>3$ and $I$ generated by the 2x2 minors. Let $M=(x_{11},x_{12},x_{13})$. I would like to be able to understand the direct summands of syzygies of $M$. They are all maximal Cohen-Macaulay, but that's all I know.

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Polynomials over what kind of object? – Ryan Budney Dec 18 '09 at 7:40
Any field will do. – Hailong Dao Dec 18 '09 at 7:54
I've given this a little thought over the years, and I think it might be very hard even to tell algorithmically whether a given module (given as a cokernel of some matrix) is indecomposable. Can Singular tell whether a (mildly) non-commutative ring such as $\operatorname{End}(M)$ is local or not? I know that M2 cannot. – Graham Leuschke Dec 18 '09 at 21:52
Good point, Graham! May be there should be a project! – Hailong Dao Dec 18 '09 at 21:56
May I draw you attention to some recent criterion of decomposability: arXiv:1305.2256 . Probably it does not help for the syzygies of this particular $M$, but it seems quite useful in many cases. – Dmitry Kerner Jul 19 '15 at 18:07

I believe you're taking the analogy with representation theory too literally. The ADE singularities have finitely many indecomposable MCM modules, not indecomposable modules. Also, if $M_i$ and $M_j$ are indecomposable MCM modules, $Hom_R(M_i,M_j)$ is going to be another reflexive module, and almost certainly not zero. (Specifically, there are certainly examples where $i\neq j$ and the $Hom$ is not zero.) The later comments in the triangulated situation are correct, but not helpful in general. – Graham Leuschke Dec 18 '09 at 14:56
Thanks, Brian. I was thinking of something like $k[x_11,...,x_33]$ modulo the 2x2 minors, so typically there will be quite a few even MCM indecomposables. – Hailong Dao Dec 18 '09 at 20:40