混合Weil上同调
Mixed Weil cohomologies
摘要：
对于正则格式$S$和给定的特征为零的域$\KK$，我们定义了光滑$S$ -格式上的$\KK$ -线性混合Weil上同调的概念，主要是：Nisnevich下降，同伦不变性，稳定性（这意味着$\GG_{m}$的上同调行为正确）和Künneth公式.我们证明了任何定义在光滑$S$ -方案上的混合Weil上同调诱导了$S$上的某些合适的三角化动机范畴到域$\KK$的导出范畴的对称monoidal实现。这意味着关于光滑和投射$S$ -格式（当$S$是完满域的谱时，可以推广到光滑$S$ -格式）的上同调的有限性定理和Poincaré对偶定理。这种形式主义也提供了一个方便的工具来理解这种上同调理论的比较。我们的主要例子是代数de Rham上同调和刚性上同调，以及与它们相关的Berthelot-Ogus同构。
英文摘要：
We define, for a regular scheme $S$ and a given field of characteristic zero $\KK$, the notion of $\KK$-linear mixed Weil cohomology on smooth $S$-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance, stability (which means that the cohomology of $\GG_{m}$ behaves correctly), and Künneth formula. We prove that any mixed Weil cohomology defined on smooth $S$-schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over $S$ to the derived category of the field $\KK$. This implies a finiteness theorem and a Poincaré duality theorem for such a cohomology with respect to smooth and projective $S$-schemes (which can be extended to smooth $S$-schemes when $S$ is the spectrum of a perfect field). This formalism also provides a convenient tool to understand the comparison of such cohomology theories. Our main examples are algebraic de Rham cohomology and rigid cohomology, and the Berthelot-Ogus isomorphism relating them.
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