Auslander-Reiten duality for subcategories

Abstract

Auslander-Reiten duality for module categories is generalized to some sufficiently nice subcategories. In particular, our consideration works for ${\mathcal{P}}^{<\infty }(\Lambda )$, the subcategory consisting of finitely generated modules with finite projective dimension over an artin algebra $\Lambda$, and also, the subcategory of Gorenstein projectove modules of $\rm{mod}\mbox{-}Λ$, denoted by $\rm{Gprj}\mbox{-}Λ$. In this paper, we give a method to compute the Auslander-Reiten translation in ${\mathcal{P}}^{<\infty }(\Lambda )$ whenever $\Lambda$ is a $1$-Gorenstein algebra. In addition, we characterize when the Auslander-Reiten translation in $\rm{Gprj}\mbox{-}Λ$ is the first syzygy and provide many algebras having such property.

arXiv commentA major revision, Title, abstract and introduction completely changed, adding some new results

References in corpus (1)

• Singularity categories of deformations of Kleinian singularities

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