${\cal H}$-cohomologies versus algebraic cycles
| dc.creator | Barbieri-Viale, Luca | |
| dc.date | 1994-08-08 | |
| dc.date.accessioned | 2026-07-07T09:06:08Z | |
| dc.date.available | 2026-07-07T09:06:08Z | |
| dc.description | Global intersection theories for smooth algebraic varieties via products in {\it appropriate}\, Poincaré duality theories are obtained. We assume given a (twisted) cohomology theory $H^*$ having a cup product structure and we let consider the ${\cal H}$-cohomology functor $X\leadsto H^{\#}_{Zar}(X,{\cal H}^*)$ where ${\cal H}^*$ is the Zariski sheaf associated to $H^*$. We show that the ${\cal H}$-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 ${\cal H}$-cohomology are obtained. We therefore obtain, for $X$ smooth, Chern classes $c_{p,i} : K_i(X) \to H^{p-i}(X,{\cal H}^p)$ from the Quillen $K$-theory to ${\cal H}$-cohomologies according with Gillet and Grothendieck. We finally obtain the ``blow-up formula'' $$H^p(X',{\cal H}^q) \cong H^p(X,{\cal H}^q)\oplus \bigoplus_{i=0}^{c-2} H^{p-1-i}(Z,{\cal H}^{q-1-i})$$ 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. | |
| dc.description | 51 pages, LaTeX 2.09 | |
| dc.identifier | https://arxiv.org/abs/alg-geom/9408002 | |
| dc.identifier | http://arxiv.org/abs/alg-geom/9408002 | |
| dc.identifier | Math. Nachr. 184 (1997), 5-57 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/149911 | |
| dc.subject | Algebraic Geometry | |
| dc.title | ${\cal H}$-cohomologies versus algebraic cycles | |
| dc.type | text |