Representation types of the category of subprojective representations of a finite poset over K[t]/(tm) and a solution of a Birkhoff type problem

Abstract

We determine the representation type (wild, tame, polynomial growth) of the category fspr(I, F m ) of filtered subprojective F m -representations of a finite poset I in terms of m and I , where F m = K[t]/(t m ), m 1, and K is an algebraically closed field. Criteria for tameness, wildness and tameness of nonpolynomial growth of fspr(I, F m ) are given in Theorems 1.1 and 1.2. As an application, a solution of Birkhoff's type problem [G. Birkhoff, Subgroups of abelian groups, Proc. London Math. Soc. 38 (1934) 385-401] for the category rep ft (I, F m ) of filtered I -chains of F m -modules is given in Section 5, by determining the representation type rep ft (I, F m ).