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.