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Let $K$ be an algebraically closed field. I'm interested in isomorphism classes of triples $(V,f,g)$ where $V$ is a finite dimensional $K$-vector space and $f,g$ are commuting endomorphisms of $V$. What I expect is a "normal form" theorem, similar to Jordan's one - which solves the case of a single endomorphism.

If we can find some smooth irreducible curve $\, \mathcal{C} \, : \, P(x,y) = 0$ defined over $K$ such that $P(f,g) = 0$, then the $K[X,Y]$-module structure on $V$ factors through the coordinate ring of $\mathcal{C}$, which is a Dedekind domain. One can then apply the "structure theorem for finitely generated modules over Dedekind domains", which yields : $(V,f,g)$ has a (essentially unique) direct sum decomposition into triples of the form $(K[X,Y]/(X,Y)^n,\alpha + X, \beta + Y)$, where $n \geq 1$ and $(\alpha,\beta)$ is some point of $\mathcal{C}$.

Is there a similar result in absence of such a nice curve ? (This question was raised in the comments of this mathstackexchange question.)

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    $\begingroup$ In general, there is no such structure theorems. But, this is a well studied case and one can at least say that there is a flat deformation with general member having the property you mention. This fails spectacularly if you take more commuting endomorphisms. $\endgroup$ – Mohan Aug 29 '13 at 18:16
  • $\begingroup$ Thanks. Could you give references for these facts ? (or at least indicate what are the relevant keywords) $\endgroup$ – js21 Aug 29 '13 at 18:56
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The classification of $K[X,Y]$-modules of finite length is of wild type, meaning that a classification of these modules would contain within it a classification of all finite-dimensional $\Lambda$-modules for all finite-dimensional $K$-algebras $\Lambda$. This is a theorem of Gelfand-Ponomarev.

In fact, even the classification of those $K[X,Y]$-modules annihilated by $(X^2, XY^2,Y^3)$ is a wild problem, as shown by Drozd.

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  • $\begingroup$ Thanks. (Note, however, that it doesn't exclude the existence of such a classification theorem ...!) Do you have a concrete example of such a module which does not decompose as in my question ? $\endgroup$ – js21 Aug 29 '13 at 19:33
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    $\begingroup$ Actually it also contains the finite dimensional rep theory for all finitely generated K-algebras and the first order theory for the fd K[x,y]-algebras is undecidable. $\endgroup$ – Benjamin Steinberg Aug 30 '13 at 0:13

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