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ℋ︁ — Cohomologies Versus Algebraic Cycles
Author(s) -
Viale Luca Barbieri
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19971840102
Subject(s) - mathematics , cohomology , pure mathematics , sheaf , de rham cohomology , cup product , sheaf cohomology , functor , intersection homology , čech cohomology , chern class , codimension , equivariant cohomology , algebra over a field
Global intersection theories for smooth algebraic varieties via products in appropriate Poincaré duality theories are obtained. We assume given a (twisted) cohomology theory H * having a cup product structure and we consider the ℋ‐ cohomology functor X ↝ H # Zar ( X, ℋ* ) where ℋ* is the Zariski sheaf associated to H *. We show that the 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 ℋ‐ cohomology are obtained. We therefore obtain, for X smooth, Chern classes c P, i : K i ( X )→ H p‐i ( X,ℋ p ) from the Quillen K‐ theory to ℋ‐ cohomologies according to Gillet and Grothendieck. We finally obtain the “blow‐up formula”where X′ 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.