Recently I became curious about the following question:


Let $V$ be a finite dimensional vector space over $k$ and let $A\_1, \cdots, A\_n: V \rightarrow V$ be a set of commuting maps. Question: describe the structure of $V$ as a module over $k[x\_1, \cdots, x\_n]$ where $x\_i$ acts by $A\_i$.


Since $V$ has a finite length as $k[x\_1, \cdots, x\_n]$-module, after we quotient by annihilator of $V$, we have a module over artinian ring (geometrically speaking, support is discrete set), so $V$ is isomorphic to direct sum of it's localizations at prime ideals (all primes are maximal in this case). Is there a description of each of the local components?
We can assume that $k$ is algebraically closed, so that Nullstellensatz might help.


Thanks for your replies. 



Generalization of Jordan Decomposition for Several Commuting Operators
It is well known that representation theory of a (even *commutative*) Artinian $k$-algebra $R$ can be wild (meaning that one can embed $\mod(A)$ into $\mod(R)$ for any finite-dimensional, not necessarily commutative, $k$-algebra $A$). The easiest example is $k[x,y]/(x^2,xy^2,y^3)$, see Example 4 in [this paper](http://arxiv.org/abs/1008.2465) and the references there. 


