$\mathcal{H}$-cohomologies versus algebraic cycles

Abstract

Global intersection theories for smooth algebraic varieties via products in {\it appropriate}\, Poincaré duality theories are obtained. We assume given a (twisted) cohomology theory $H^{\ast }$ having a cup product structure and we let consider the $\mathcal{H}$-cohomology functor $X⇝H_{Zar}^{\#}(X,{\mathcal{H}}^{\ast })$ where ${\mathcal{H}}^{\ast }$ is the Zariski sheaf associated to $H^{\ast }$. We show that the $\mathcal{H}$-cohomology rings generalize the classical ``intersection rings'' obtained via rational or algebraic equivalences. Several basic properties e.g.\, Gysin maps, projection formula and projective bundle decomposition, of $\mathcal{H}$-cohomology are obtained. We therefore obtain, for $X$ smooth, Chern classes $c_{p,i}:K_i(X)\to H^{p-i}(X,{\mathcal{H}}^p)$ from the Quillen $K$-theory to $\mathcal{H}$-cohomologies according with Gillet and Grothendieck. We finally obtain the ``blow-up formula'' $$H^p(X^{\prime },{\mathcal{H}}^q)\cong H^p(X,{\mathcal{H}}^q)\oplus \underset{i=0}{\overset{c-2}{⨁}}{H}^{p-1-i}(Z,{\mathcal{H}}^{q-1-i})$$ where $X^{\prime }$ is the blow-up of $X$ smooth, along a closed smooth subset $Z$ of pure codimension $c$. Singular cohomology of associated analityc space, étale cohomology, de Rham and Deligne-Beilinson cohomologies are examples for this setting.

51 pages, LaTeX 2.09

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