Monomorphism categories, cotilting theory, and Gorenstein-projective modules

Abstract

The monomorphism category ${\mathcal{S}}_n(\mathcal{X})$ is introduced, where $\mathcal{X}$ is a full subcategory of the module category $A$-mod of Artin algebra $A$. The key result is a reciprocity of the monomorphism operator ${\mathcal{S}}_n$ and the left perpendicular operator ${}^{\perp }$: for a cotilting $A$-module $T$, there is a canonical construction of a cotilting $T_n(A)$-module $\mathbf{m}(T)$, such that ${\mathcal{S}}_n{(}^{\perp }T)={}^{\perp }\mathbf{m}(T)$. As applications, ${\mathcal{S}}_n(\mathcal{X})$ is a resolving contravariantly finite subcategory in $T_n(A)$-mod with $\widehat{{\mathcal{S}}_n(\mathcal{X})}=T_n(A)$-mod if and only if $\mathcal{X}$ is a resolving contravariantly finite subcategory in $A$-mod with $\hat{\mathcal{X}}=A$-mod. For a Gorenstein algebra $A$, the category $T_n(A)\mbox{-}\mathcal Gproj$ of Gorenstein-projective $T_n(A)$-modules can be explicitly determined as ${\mathcal{S}}_n{(}^{\perp }A)$. Also, self-injective algebras $A$ can be characterized by the property $T_n(A)\mbox{-}\mathcal Gproj = \mathcal S_n(A)$. Using ${\mathcal{S}}_n(A)={}^{\perp }\mathbf{m}(D(A_A))$, a characterization of ${\mathcal{S}}_n(A)$ of finite type is obtained.

20 pages