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 $\rm{ mod}\mbox{ -}\Lambda$, denoted by $\rm{ Gprj}\mbox{ -}\Lambda$. 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 $\rm{ Gprj}\mbox{ -}\Lambda$ is the first syzygy and provide many algebras having such property.
Categories: math.RT