On fully wild categories of representations of posets

Abstract

Assume that \(I\) is a finite partially ordered set and \(k\) is a field. We prove that if the category \(\mbox{ prin}(kI)\) of prinjective modules over the incidence \(k\)-algebra \(kI\) of \(I\) is fully \(k\)-wild then the category \({\bf fpr}(I,k)\) of finite dimensional \(k\)-representations of \(I\) is also fully \(k\)-wild. A key argument is a construction of fully faithful exact endofunctors of the category of finite dimensional \(k\langle x,y\rangle\)-modules, with the image contained in certain subcategories.