Some remarks of Hochschild homology and semi-orthogonal decompositions
Some remarks of Hochschild homology and semi-orthogonal decompositions
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Hochschild同调和半正交分解的一些注记
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发表时间：
2021
期刊：
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通讯作者：
Xun Lin
中科院分区：
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作者：
Xun Lin
Kotaro Kawatani和Shinnosuke Okawa证明了，给定一个非平凡的半正交分解Perf（X）= λ A，B λ，并假设ωX的基轨迹是一个真闭子集，所有的摩天大楼层k（x）（x ∈ Bs）|ωX|只属于其中一种成分很自然会问它是哪一个，以及我们是否可以通过某些线性不变量来确定它。在本文中，我们利用有支撑的相干层的导出范畴的Hochschild同调，提供了另一个证明：如果一个分支的−n Hochschild同调非零，那么我们上面考虑的摩天大楼层属于这样的分支，这最初是由Dmitrii Pirozhkov [29，引理5.3]证明的。进一步证明了Kuznetsov提出的关于Perf（X）（dimX = n）的n-Calabi-Yau可容许子范畴的分类的猜想.最后，我们注意到具有支撑的导出范畴的加法不变量可以为半正交分解提供更多的线性障碍。
Given a nontrivial semi-orthogonal decomposition Perf(X) = 〈A,B〉, and assume that the base locus of ωX is a proper closed subset, it was proved by Kotaro Kawatani and Shinnosuke Okawa that all skyscraper sheaves k(x) with x / ∈ Bs|ωX| 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 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 [29, Lemma 5.3]. Furthermore, we prove a conjecture proposed by Kuznetsov about classifying n-Calabi-Yau admissible subcategory of Perf(X) (dimX = 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.