We prove a product formula for the determinant of the cohomology of an étale sheaf with $\ell $ -adic coefficients over an arbitrary proper scheme over a perfect field of positive characteristic p distinct from $\ell $ . The local contributions are constructed by iterating vanishing cycle functors as well as certain exact additive functors that can be considered as linearised versions of Artin conductors and local $\varepsilon $ -factors. We provide several applications of our higher dimensional product formula, such as twist formulas for global $\varepsilon $ -factors.

中文翻译：

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更高维度的几何局部因素

我们证明了具有$\ell$ 进系数的 étale 层的上同调行列式的乘积公式，在任意适当方案上，在一个不同于 $\ell$ 的正特征p的完美域上。局部贡献是通过迭代消失循环函子以及某些精确加法函子构建的，这些函子可以被视为 Artin 导体和局部 $\varepsilon $ 因子的线性化版本。我们提供了高维乘积公式的几种应用，例如全局 $\varepsilon $ 因子的扭曲公式。