Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture
Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture
复制标题
刚性几何、Lefschetz-Verdier 迹公式和德利涅猜想
DOI：
10.1007/s002220050129
复制
发表时间：
1997
影响因子：
3.1
通讯作者：
K. Fujiwara
中科院分区：
文献类型：
--
作者：
K. Fujiwara
在抽象代数几何中，Lefschetz-Verdier迹公式表示由对应在上同调上诱导的自同态的整体迹，作为在不动点附近的局部贡献之和，称为局部项。虽然该公式具有相当普遍的适用性，但局部项的显式计算很难进行，并且没有好的公式可以用于一般对应。在复数上，一个有用的公式是已知的一个很好的类称为弱双曲[G-M]。另一方面，在正特征线中，P. Deligne指出，如果我们用Frobenius的高次幂构成一个给定的对应，局部项将大大简化。取有限域Fq的代数闭包k上的真簇X，对应a：Y ~ XxkX，X的开集U上的光滑QrK，其中开集U由a固定（用j表示包含，{在k中可逆）。然后
In abstract algebraic geometry, the Lefschetz-Verdier trace formula expresses the global trace of an endomorphism induced on cohomology by a correspondence as a sum of local contributions, called local terms, near fixed points. Though the formula holds quite generally, the explicit calculation of local terms is very hard to carry out, and no good formula is known for general correspondences. Over the complex numbers a useful formula is known for a good class called weakly hyperbolic [G-M]. On the other hand, in positive characteristics, P. Deligne conjectured that the local terms would be much simplified if we composed a given correspondence with a high power of Frobenius. Take a proper variety X over an algebraic closure k of a finite field Fq, a correspondence a : Y ~ X xk X, and a smooth Qr K on an open set U of X fixed by a (denote the inclusion by j , and { is invertible in k). Then