Timeline for structure theorem for modules

7 events

when toggle format	what		by	license	comment
Jun 20, 2011 at 16:45	vote	accept	Anil P
Jun 14, 2011 at 21:08	comment	added	Mariano Suárez-Álvarez		A reference for the result on $k[x,y]$ is [I. M. Gelfand and V. A. Ponomarev, Remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space, Functional Anal. Appl. 3 (1969) 325-326.] They prove that classifying artinian modules over $k[x,y]$ is as hard as classifying modules over a free algebra on three generators. Drozd's work implies this last problem is wild.
Jun 14, 2011 at 20:46	history	edited	Yemon Choi		added better tags
Jun 14, 2011 at 16:08	comment	added	Graham Leuschke		To add to Donu's comment in the spirit of unknown's answer: the finite-length modules over $k[x,y]$ are of wild representation type -- they can't be parametrized by the points of any finite-dimensional variety. Put another way, a classification of those modules would entail a classification of \emph{all} modules over \emph{every} finite-dimensional $k$-algebra. So: hopeless. The result about $k[x,y]$ is due to Drozd, I think.
Jun 14, 2011 at 15:40	answer	added	user9072		timeline score: 6
Jun 14, 2011 at 15:14	comment	added	Donu Arapura		I'm guessing you're referring to the structure of modules over a principal ideal domain? A complete classification of modules over a polynomial ring in $2$ variables is already very difficult, but a lot is known. What sort of answer were you hoping for?
Jun 14, 2011 at 14:37	history	asked	Anil P	CC BY-SA 3.0