Recollements from generalized tilting

Abstract

Let A \mathcal {A} be a small dg category over a field k k and let U \mathcal {U} be a small full subcategory of the derived category D A \mathcal {D}\mathcal {A} which generates all free dg A \mathcal {A} -modules. Let ( B , X ) (\mathcal {B},X) be a standard lift of U \mathcal {U} . We show that there is a recollement such that its middle term is D B \mathcal {D}\mathcal {B} , its right term is D A \mathcal {D}\mathcal {A} , and the three functors on its right side are constructed from X X . This applies to the pair ( A , T ) (A,T) , where A A is a k k -algebra and T T is a good n n -tilting module, and we obtain a result of Bazzoni–Mantese–Tonolo. This also applies to the pair ( A , U ) (\mathcal {A}, \mathcal {U}) , where A \mathcal {A} is an augmented dg category and U \mathcal {U} is the category of ‘simple’ modules; e.g., A \mathcal {A} is a finite-dimensional algebra or the Kontsevich–Soibelman A ∞ A_\infty -category associated to a quiver with potential.