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Let S(n) be the category of invariant subspaces of nilpotent operators withnilpotency index at most n. Such submodule categories have been studied alreadyin 1936 by Birkhoff, they have attracted a lot of attention in recent years,for example in connection with some weighted projective lines (Kussin, Lenzing,Meltzer). On the other hand, we consider the preprojective algebra of type A_n;the preprojective algebras were introduced by Gelfand and Ponomarev, they arenow of great interest, for example they form an important tool to study quantumgroups (Lusztig) or cluster algebras (Geiss, Leclerc, Schroeer). Directconnections between the submodule category S(n) and the module category of thepreprojective algebra of type A_{n-1} have been established quite a long timeago by Auslander and Reiten, and recently also by Li and Zhang, but apparentlythis remained unnoticed. The aim of this note is to provide details on thisrelationship. As a byproduct we see that here we deal with ideals I intriangulated categories T such that I is generated by an idempotent and T/I isabelian.

From submodule categories to preprojective algebras