Auslander-Reiten duality for subcategories

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