Recollements from generalized tilting

Let $\ca$ be a small dg category over a field $k$ and let $\cu$ be a small full subcategory of the derived category $\cd\ca$ which generate all free dg $\ca$-modules. Let $(\cb,X)$ be a standard lift of $\cu$. We show that there is a recollement such that its middle term is $\cd\cb$, its right term is $\cd\ca$, 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 $(\ca,\cu)$, where $\ca$ is an augmented dg category and $\cu$ is the category of `simple’ modules, e.g. $\ca$ is a finite-dimensional algebra or the Kontsevich–Soibelman $A_\infty$-category associated to a quiver with potential.