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摘要翻译：
对于一个正则格式$S$和一个给定的特征为零的域$\kk$,我们通过一组简单的性质定义了光滑$S$-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k“unneth公式。我们证明了定义在光滑的$S$-格式上的任何混合Weil上同调将$S$上的某些合适的三角化动机范畴的对称单向实现到域$\kk$的导出范畴。这意味着这样一个上同调关于光滑射影$S$-格式（当$S$是完美场的谱时，可以推广到光滑$S$-格式）的有限性定理和Poincar对偶定理。这种形式主义也为理解此类上同调理论的比较提供了方便的工具。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。
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英文标题：
《Mixed Weil cohomologies》
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作者：
Denis-Charles Cisinski, Fr\'ed\'eric D\'eglise
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最新提交年份：
2009
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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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 K\"unneth 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 Poincar\'e 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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PDF链接：
https://arxiv.org/pdf/0712.3291
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