Gorenstein projective objects in comma categories

Abstract

Let $\mathcal {A}$ and $\mathcal {B}$ be abelian categories and ${\mathbf {F}} :\mathcal {A}\rightarrow \mathcal {B}$ an additive and right exact functor which is perfect, and let $({\mathbf {F}},\mathcal {B})$ be the left comma category. We give an equivalent characterization of Gorenstein projective objects in $({\mathbf {F}},\mathcal {B})$ in terms of Gorenstein projective objects in $\mathcal {B}$ and $\mathcal {A}$. We prove that there exists a left recollement of the stable category of the subcategory of $({\mathbf {F}},\mathcal {B})$ consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in $\mathcal {B}$ and $\mathcal {A}$. Moreover, this left recollement can be filled into a recollement when $\mathcal {B}$ is Gorenstein and ${\mathbf {F}}$ preserves projectives.