2015/10/17 by Luo, Xiu-Hua, Zhang, Pu
#16E65 #16G10 #16G50 #16G60 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1510.05124
For a k-algebra A, a quiver Q, and an ideal I of kQ generated by monomial relations, let Λ: = A⊗k kQ/I. We introduce the monic representations of (Q, I) over A. We give properties of the structural maps of monic representations, and prove that the category \rm mon(Q, I, A) of the monic representations of (Q, I) over A is a resolving subcategory of \rm rep(Q, I, A). We introduce the condition \rm(G). The main result claims that a \m-module is Gorenstein-projective if and only if it is a monic module satisfying \rm(G). As consequences, the monic \m-modules are exactly the projective \m-modules if and only if A is semisimple; and they are exactly the Gorenstein-projective \m-modules if and only if A is selfinjective, and if and only if \rm mon(Q, I, A) is Frobenius.