# Moduli space of modules with fixed length

Let $R$ be a (commutative) local Artinian ring, with an algebraically closed residue field $k$. I am interested in the set $L_n(R)$ of isomorphism classes of $R$-modules of length $n$.

If $R$ is a $k$-algebra, this is a very classical subject. The set $\text{mod}_R^n$ of $R$-module structures on a fixed $k$-vector space of dimension $n$ has a natural structure of an affine $k$-scheme, and $L_n(R)$ can be viewed as the set of orbits of $GL_n(k)$ on $\text{mod}_R^n$. This has been exploited very frequently in the representation theory of Artin algebras.

I feel like the following question would be known to the right people, but could not find a satisfying reference:

Has the set $L_n(R)$ been studied in this general setting? I am more interested in geometric approaches, but any references would be welcome.

(The only remotely relevant reference I could find is this paper by Bjorn Poonen on moduli space of commutative algebras of finite rank over a commutative base).

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Long, I'm sure I have no hope to answer this question but I'm curious as to which artinian rings are your motivating examples. For example, you are considering an artinian local ring $R$ with algebraically closed residue field $k$ but which is not a $k$-algebra, right? What's the canonical example? –  Karl Schwede Sep 10 '10 at 1:36
@Karl: I am interested in artinian rings in general, but the questions I like to answer to is unaffected by making a faithfully flat extension, so you can assume alg. closed residue field for free. Sometimes it helps. Also, assuming alg. closed from the beginning made my second paragraph simpler. –  Hailong Dao Sep 10 '10 at 2:58