Semi-orthogonal decompositions over singular schemes

Question

Where can I find any more or less explicit semi-orthogonal decompositions of derived categories of perfect complexes or of bounded derived categories for singular schemes that are proper over a ring R?

I guess that one can take a "Beilinson-type" decomposition of $D^{perf}(\mathbb{P}^n(\operatorname{Spec}\,R))$ where $R$ is not regular; one can probably take the "inverse image" of this decomposition to something like $D^{perf}(\mathbb{P}^n\times V(\operatorname{Spec}\,R))$. Yet I would like to have some references for this argument. Moreover, I would like to find some "more interesting" examples. This question is certainly related to examples of explicit descriptions of derived categories of singular varieties.

Answer 1

There is a notion of base change for semiorthogonal decompositions, applying which you can induce semiorthogonal decompositions over rings from those over fields, see Kuznetsov, Alexander. Base change for semiorthogonal decompositions. Compos. Math. 147 (2011), no. 3, 852--876. You can also find nontrivial examples of semiorthofonal decompositions for singular toric surfaces in https://arxiv.org/abs/1809.10628.