Recollements from generalized tilting

Abstract:

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