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Weil cohomology theories and their motivic Hopf algebroids
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In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the theory of motives, we give three equivalent viewpoints on Weil cohomology theories: as a cohomology theory on smooth varieties, as a motivic spectrum and as a realization functor. We also associate to every Weil cohomology theory a motivic Hopf algebroid generalizing a previous construction of the author for the Betti cohomology. The main result proven in the paper is the connectivity of the motivic Hopf algebroids associated to the classical Weil cohomology theories.
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Cited by 1 Pith paper
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A motivic approach to rational $p$-adic cohomologies
Smooth complete intersections in projective toric varieties over p-adic fields satisfy the p-adic weight-monodromy conjecture, here re-proven through motivic nearby cycles and a monodromy-category equivalence.
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