2007/12/31 by Denis-Charles Cisinski, Frédéric Déglise · 2 citations
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Cohomology #Cup product #De Rham cohomology #Descent (aeronautics) #Equivariant cohomology #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy category #Mathematics #Motivic cohomology #Pure mathematics #math.AG #msc:14F30 #msc:14F40 #msc:14F42 #msc:18E30 #msc:19E15 #msc:55N40 #msc:55U25 #msc:55U30
paper · pdf · doi:10.1016/j.aim.2011.10.021
published as Advances in Mathematics 230 (2012), no. 1, 55-130 · update references; hopefully improve the exposition
arxiv created 2009/12/10 · openalex publication_date 2011/12/15 · arxiv updated 2012/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 \GGm 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.