H-COHOMOLOGIES VERSUS ALGEBRAIC CYCLES

Global intersection theories for smooth algebraic varieties via produc ts in appropriate Poincare duality theories are obtained. We assume gi ven a (twisted) cohomology theory H having a cup product structure an d we consider the H-cohomology functor X similar to H-Zar(#)(X,H) wher e H is the Zariski sheaf associated to H*. We show that the H-cohomol ogy rings generalize the classical ''intersection rings'' obtained via rational or algebraic equivalences. Several basic properties e.g. Gys in maps, projection formula and projective bundle decomposition, of H- cohomology are obtained. We therefore obtain, for X smooth, Chern clas ses c(p,i) : K-i(X) --> Hp-i(X, H-p) from the Quillen K - theory to H- cohomologies according to GILLET and GROTHENDIECK. We finally obtain t he ''blow- up formula'' H-p(X', H-q) congruent to H-p(X,H-q) + (i-0)+H -c-2(p-1-i)(Z,Hq-1-i) where X' is the blow-up of X' smooth, along a cl osed smooth subset Z of pure codimension c. Singular cohomology of ass ociated analityc space, etale cohomology, de Rham and Deligne-Beilinso n cohomologies are examples for this setting.