Absolute cohomological purity

— 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 AtiyahHirzebruch 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. 1. The standard statement of GROTHENDIECK'S purity conjecture is: CONJECTURE 1.1 ([16], 13.1.4). Let X be a regular prescheme, X ' ^ X an open subscheme, and i: T -+ X' a closed immersion. Suppose that V is regular, and that T^X has codimension d at each point. Let / be a prime number invertible in (9^. Then the local cohomology sheaves are given by (1.1): r 0, f^2d, ( 1 . 1 ) ^^Z/n-^^^ ,=2d. (*) Partially supported by NSF grant MCS-8108814(A01). Texte rccu Ie 14 janvier 1984, revise Ie 26 avril 1984. R. W. THOMASON, Department of Mathematics, Johns Hopkins University, Baltimore, Md 21218, U.S.A. BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE 0037-9434/1984/03 397 10/S 3.00 © Gauthier-Villars