Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some remarks of Hochschild homology and semi-orthogonal decompositions   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.

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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: astro-ph
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The categories of this paper are: econ
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th