Monic modules and semi-Gorenstein-projective modules

Abstract

The category gp ( Λ ) of Gorenstein-projective modules over tensor algebra Λ = A ⊗ k B can be described as the monomorphism category mon ( B , gp ( A ) ) of B over gp ( A ) . In particular, Gorenstein-projective Λ-modules are monic. In this paper, we find the similar relation between semi-Gorenstein-projective Λ-modules and A -modules, via monic modules, namely, mon ( B , A ⊥ ) = mon ( B , A ) ∩ Λ ⊥ . Using this, it is proved that if A is weakly Gorenstein, then Λ is weakly Gorenstein if and only each semi-Gorenstein-projective Λ-modules are monic; and that if B = k Q with Q a finite acyclic quiver, then Λ is weakly Gorenstein if and only if A is weakly Gorenstein. However, this relation itself does not answer the question whether there exist double semi-Gorenstein-projective Λ-modules which are not monic. Using the recent discovered examples of double semi-Gorenstein-projective A -modules which are not torsionless, we positively answer this question, by explicitly constructing a class of double semi-Gorenstein-projective T 2 ( A ) -modules with one parameter such that they are not monic, and hence not torsionless. The corresponding results are obtained also for the monic modules and semi-Gorenstein-projective modules over the triangular matrix algebras given by bimodules .