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.