Vol. 264, No. 1, 2013

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 293: 1
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Vol. 286: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Monic representations and Gorenstein-projective modules

Xiu-Hua Luo and Pu Zhang

Vol. 264 (2013), No. 1, 163–194
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
Authors
Xiu-Hua Luo
Department of Mathematics
Shanghai Jiao Tong University
Dongchuan Road 800
Shanghai, 200240
China
Pu Zhang
Department of Mathematics
Shanghai Jiao Tong University
Dongchuan Road 800
Shanghai, 200240
China