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 ${\rm \bf m}(T)$, such that $\mathcal S_n(^\perp T) = \ ^\perp {\rm \bf m}(T)$. As applications, $\mathcal S_n(\mathcal X)$ is a resolving contravariantly finite subcategory in $T_n(A)$-mod with $\hat{\mathcal S_n(\mathcal X)} = T_n(A)$-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 $\mathcal S_n(A)= \ ^\perp {\rm \bf m}(D(A_A))$, a characterization of $\mathcal S_n(A)$ of finite type is obtained.