Monomorphism categories, cotilting theory, and Gorenstein-projective modules

Abstract

The monomorphism category S n ( X ) is introduced, where X is a full subcategory of the module category A-mod of an Artin algebra A. The key result is a reciprocity of the monomorphism operator S n and the left perpendicular operator ⊥: for a cotilting A-module T, there is a canonical construction of a cotilting module m ( T ) over the upper triangular matrix algebra T n ( A ) , such that S n ( T ⊥ ) = m ⊥ ( T ) . As applications, S n ( X ) is a resolving contravariantly finite subcategory in T n ( A ) -mod with S n ( X ) ˆ = T n ( A ) -mod if and only if X is a resolving contravariantly finite subcategory in A-mod with X ˆ = A -mod. For a Gorenstein algebra A, the category T n ( A ) - G proj of Gorenstein-projective T n ( A ) -modules can be explicitly determined as S n ( A ⊥ ) . Also, self-injective algebras A can be characterized by the property T n ( A ) - G proj = S n ( A ) . Finally, we obtain a characterization of those categories S n ( A ) which have finite representation type in terms of Auslanderʼs representation dimension.