Hochschild (co-)homology of schemes with tilting object
Hochschild (co-)homology of schemes with tilting object
复制标题
倾斜物体方案的 Hochschild（共）同调
DOI：
10.1090/s0002-9947-2012-05577-2
复制
发表时间：
2010
影响因子：
1.3
通讯作者：
L. Hille
中科院分区：
文献类型：
--
作者：
R. Buchweitz;L. Hille
给定一个k-概型X，其中T是倾斜的，本文证明了X的Hochschild（余）同调同构于A= End_{X}（T）的Hochschild（余）同调.我们对待更一般的相对情况下，当$X$是平坦的仿射计划$Y=\Spec R$和倾斜对象满足一个适当的托尔独立条件超过$R$。在应用中，我们发现$X$ over $Y$的Hochschild同调在负度中为零，$X$ over $Y$的光滑性等价于$A$ over $R$的光滑性，对于$X$的光滑射影格式，我们得到Hochschild同调集中在零度.利用Hochschild同调在特征零点的Hodge分解，对于$X$在$Y$上光滑，Hodge群$H^{q}（X，\Omega_{X/Y}^{p}）$对于$p < q$为零，而在绝对情况下，它们甚至对于$p\neq q$为零。我们举例说明了商奇点的crepant决议的结果，特别是对射影空间上的标准丛的全空间。
Given a $k$--scheme $X$ that admits a tilting object $T$, we prove that the Hochschild (co-)homology of $X$ is isomorphic to that of $A= End_{X}(T)$. We treat more generally the relative case when $X$ is flat over an affine scheme $Y=\Spec R$ and the tilting object satisfies an appropriate Tor-independence condition over $R$. Among applications, Hochschild homology of $X$ over $Y$ is seen to vanish in negative degrees, smoothness of $X$ over $Y$ is shown to be equivalent to that of $A$ over $R$, and for $X$ a smooth projective scheme we obtain that Hochschild homology is concentrated in degree zero. Using the Hodge decomposition \cite{BFl2} of Hochschild homology in characteristic zero, for $X$ smooth over $Y$ the Hodge groups $H^{q}(X,\Omega_{X/Y}^{p})$ vanish for $p < q$, while in the absolute case they even vanish for $p\neq q$. We illustrate the results for crepant resolutions of quotient singularities, in particular for the total space of the canonical bundle on projective space.