A Krause-type recollement for Q-shaped derived categories

Abstract

Let k be a commutative noetherian hereditary ring and let Q be a small, Hom-finite, locally bounded k-linear category admitting a Serre functor and satisfying the strong retraction property, with nilpotent pseudo-radical. For every k-algebra A, we construct a recollement of the Q-shaped homo?topy category KQ(A Inj) of objectwise injective Q-shaped modules by its full subcategory Kac Q (A Inj) of exact objects and the Q-shaped derived category DQ(A). The construction uses complete hereditary cotorsion pairs and does not require A to be noetherian. If A is left noetherian, both KQ(A Inj) and Kac Q (A Inj) are compactly generated by special preenvelopes of finitely generat?ed stalk modules. We identify (Kac Q (A Inj))c with the idempotent completion of (KQ(A Inj))c/LI ((DQ(A))c ), where LI is the fully faithful left adjoint of the localization. This paper builds on and generalizes works of Krause and Liu