Some remarks of Hochschild homology and semi-orthogonal decompositions

Abstract

Given a nontrivial semi-orthogonal decomposition $\Perf(\X)=\langle \mathcal{A},\mathcal{B}\rangle$, and assume that the base locus of $\omega_{\X}$ is a proper closed subset, it was proved by Kotaro Kawatani and Shinnosuke Okawa that all skyscraper sheaves $\k(\x)$ with $\x\notin \mathsf{Bs}\vert \omega_{\X}\vert$ belong to exactly one and only one of the components. It is natural to ask which one it is, and whether we can determine this by certain linear invariants. In this note we use Hochschild homology of derived category of coherent sheaves with support to provide another proof that if the $-\n^{\th}$ Hochschild homology of a component is nonzero, then the skyscraper sheaves we consider above belong to such component, which was originally proved by Dmitrii Pirozhkov \cite[Lemma 5.3]{pirozhkov2020admissible}. Furthermore, we prove a conjecture proposed by Kuznetsov about classifying $\n$-Calabi-Yau admissible subcategory of $\Perf(\X)$ ($\dim \X=\n$) for certain projective smooth variety $\X$ if we put more assumptions to the Calabi-Yau categories. Finally we remark that the additive invariants of derived category with support could provide more linear obstructions to semi-orthogonal decompositions.