Monic representations and Gorenstein-projective modules

Xiu-Hua Luo and Pu Zhang

Vol. 264 (2013), No. 1, 163–194

DOI: 10.2140/pjm.2013.264.163

Abstract

Given a finite-dimensional algebra A over a field k, and a finite acyclic quiver Q, let Λ = A ⊗_kkQ, where kQ is the path algebra of Q over k. Then the category Λ-mod of Λ-modules is equivalent to the category Rep(Q,A) of representations of Q over A. This yields the notion of monic representations of Q over A. We denote the full subcategory of Rep(Q,A) consisting of monic representations of Q over A by Mon(Q,A). It is proved that Mon(Q,A) has Auslander–Reiten sequences.

The main result of this paper explicitly describes the Gorenstein-projective Λ-modules via the monic representations plus an extra condition. As a corollary, we prove the equivalence of three conditions: A is self-injective; Gorenstein-projective Λ-modules are exactly the monic representations of Q over A; Mon(Q,A) is a Frobenius category.

Keywords

representations of a quiver over an algebra, monic representations, Gorenstein-projective modules

Mathematical Subject Classification 2010

Primary: 16G10

Secondary: 16E65, 16G50, 16G60

Milestones

Received: 10 May 2012

Accepted: 4 February 2013

Published: 5 July 2013