A question on the Krull–Schmidt theorem The Krull–Schmidt theorem states that if $M$ is a module of a ring of finite length (equivalently, satisfies the ACC and DCC), then there is an essentially unique decomposition of $M$ as a direct sum of indecomposable modules. Moreover, such a module is indecomposable iff its endomorphism ring is local.
In general, it seems to be known that both the ACC and the DCC are necessary. Now, I am thinking about graded $k[x_1,\dotsc,x_n]$-modules, where $k$ is a field. I would assume that it might be possible to strenghen the statement. Specifically, I'd like to know:
Is it possible to drop the DCC in the context of graded $k[x_1,\dotsc,x_n]$-modules?
 A: Yes.  First, the existence of such a decomposition requires only either the ACC or the DCC.  Then, the proof of uniqueness uses the assumption that $M$ has finite length only to conclude that each indecomposable summand of $M$ has local endomorphism ring.  So it would suffice to prove that if $M$ is a finitely generated indecomposable graded $k[x_1,\dots,x_n]$-module, then its ring of (grading-preserving) endomorphisms is local.
To prove this, pick a finite set of homogeneous generators for $M$, and then a finite set of homogeneous relations between these generators that gives a presentation of $M$ as a module.  Let $N$ be the largest degree of any of the generators or relations.  Note that then $\operatorname{End}(M)$ is isomorphic to $\operatorname{End}(M/\bigoplus_{n> N} M_n)$, since an endomorphism of $M$ is given by a choice of where to send the generators which preserves all the relations, and all this data is contained in degress $\leq N$.  In particular, then, $\operatorname{End}(M/\bigoplus_{n>N} M_n)$ has no nontrivial idempotents since $M$ is indecomposable, and so $M/\bigoplus_{n>N} M_n$ is also indecomposable.  But $M/\bigoplus_{n> N} M_n$ is finite-dimensional over $k$ and thus has finite length, and so its endomorphism ring is local.  Thus $\operatorname{End}(M)$ is local, as desired.
