Monomorphism Operator and Perpendicular Operator

Abstract

For a quiver Q, a k-algebra A, and a full subcategory X of A-mod, the monomorphism category Mon(Q, X ) is introduced. The main result says that if T is an A-module such that there is an exact sequenceup to multiplicities of indecomposable direct summands, such that Mon(Q, ⊥ T ) = ⊥ (kQ ⊗ k T ).As applications, the category of the Gorenstein-projective (kQ ⊗ k A)-modules is characterized as Mon(Q, GP(A)) if A is Gorenstein; the contravariantly finiteness of Mon(Q, X ) can be described; and a sufficient and necessary condition for Mon(Q, A) being of finite type is given.

Injective objects of monomorphism categories

“…It is a full subcategory of category Rep(Q, A) consisting of all the representations of Q over A (see Section 2 below). The monomorphism category attracts much attention in recent papers, such as [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. In [6], the Gorenstein projective modules were constructed by monic representations.…”