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Let $k$ be a field, and $k[x,y]$ be the polynomial ring in two variables. Feel free to assume $k$ is algebraically closed.

  1. Are there any general structural results about finitely generated infinite-dimensional modules over $k[x,y]$?

  2. Let $(\mathbb{N}, +, 0)$ be the commutative monoid of nonnegative integers under addition. Any set $S$ equipped with a monoid action of $\mathbb{N}^2$ gives rise to a module $kS$ over the monoid ring $k[\mathbb{N}^2]\simeq k[x,y]$. In my case, $S$ is infinite and finitely generated under the action. Are there any structural results about these modules $kS$?

  3. Endow $\mathbb{N}^2$ with an $\mathbb{Z}^2$-grading, given by the monoid embedding $\mathbb{N}^2\to \mathbb{Z}^2$ which sends $(1,0)\mapsto (-1,-1)$ and $(0,1)\mapsto (1,-1)$. Let $S$ be an $\mathbb{Z}^2$-graded set with an action by the graded monoid $\mathbb{N}^2$. Repeat question (2) for the graded modules $kS$.

The setup in question (3) is the exact situation I'm facing, but answers to any of these questions are greatly appreciated. Thank you!

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    $\begingroup$ Regarding 1: classification of directly indecomposable finite length modules is already impossible (I. e. it is a wild type problem: it is as hard as classifying finite dimensional modules over a free algebra up to isomorphism). This is Gelfand-Ponomarev paper from 1969. $\endgroup$ Commented Aug 15 at 10:08
  • $\begingroup$ @DenisT Thank you for the reference! $\endgroup$ Commented Sep 17 at 19:43

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