Absolute cohomological purity

Abstract

RESUME.-Cette note demontrc la conjecture de purcte cohomologique absolve de GRO-THENDIECK pour la cohomologie etale J-adique a coefficients Q7, et mcmc pour la cohomologie etale a coefficients Z/F si / est grand.Les ingredients phncipaux dans la preuve sont Ie theoreme de localisation pour la K-theorie algebriquc, et Ie theoreme de comparaison entre la K-theorie algebrique et la K-theoric topologique.ABSTRACT.-GROTHENDIECK'S absolute cohomological purity conjecture is proved for Q7-etale cohomology, and for etale cohomology with 7.1 f coefficients if / is large.The proof depends on Quillen's localization theorem for algebraic K-thcory and my comparison of algebraic and topological X-thcory.In this note, I show how my theorem relating algebraic and topological K-theory yields an absolute cohomological purity theorem for topological X-theory as the analogue of Quillen's localization theorem for algebraic ^-theory.Under various conditions the degeneration of the Atiyah-Hirzebruch spectral sequence allows one to deduce various purity results for etale cohomology.In particular, there are very general purity theorems for Q7 -cohomology in paragraph 3.

Locations

• Bulletin de la Société mathématique de France - View - PDF