Height of exceptional collections and Hochschild cohomology of quasiphantom categories
Height of exceptional collections and Hochschild cohomology of quasiphantom categories
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异常集合的高度和准幻影范畴的 Hochschild 上同调
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发表时间：
2012
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通讯作者：
A. Kuznetsov
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作者：
A. Kuznetsov
我们定义了光滑射影簇 X 上相干滑轮派生范畴的可接受子范畴的正态霍克希尔德上同调，该分级向量空间控制从 X 的霍克希尔德上同调到该可接受子范畴的正交补的霍克希尔德上同调的限制态射。当子类别由异常集合生成时，我们定义一个新的不变量（高度），并表明在平滑射影簇 X 的派生范畴中，与高度 h 的异常集合正交，在直至 h 2 的度数上具有与 X 相同的 Hochschild 上同调。我们用它来描述一般类型的某些表面的派生类别中准幻影范畴的第二 Hochschild 上同调。我们还给出了一个特殊集合在其高度和正常霍克希尔德上同调方面的充分性的必要和充分条件。
We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective varietyX, a graded vector space which controls the restriction morphism from the Hochschild cohomology of X to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is generated by an exceptional collection, we define a new invariant (the height) and show that the orthogonal to an exceptional collection of height h in the derived category of a smooth projective varietyX has the same Hochschild cohomology asX in degrees up to h 2. We use this to describe the second Hochschild cohomology of quasiphantom categories in the derived categories of some surfaces of general type. We also give necessary and sufficient conditions for the fullness of an exceptional collection in terms of its height and of its normal Hochschild cohomology.