Gersten’s conjecture for some complexes of vanishing cycles. (English) Zbl 0827.19002

S. Bloch and A. Ogus noticed, on the second half of page 190 of their paper “Gersten’s conjecture and the homology of schemes” [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 181-201 (1974; Zbl 0307.14008)], that the proof of their main theorem (which is a cohomological version of Gersten’s conjecture in \(K\)-theory; for the latter see the Proceedings of the Summer 1972 Seattle Conference [S. M. Gersten, Lect. Notes Math. 341, 211-243 (1973; Zbl 0289.18011)]) follows from the vanishing of a certain map – in fact this reduction was discovered by Quillen in his proof, “in some important equi-characteristic cases”, of Gersten’s conjecture [D. Quillen, Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)]. – In the paper under review a strengthened version of the Bloch-Ogus observation is proved and also, therefore, a cohomological version of Gersten’s conjecture. The author remarks that his proof – which is too technical to be described here – “may apply to \(p\)-adic vanishing cycles and theories satisfying the same formalism”.


19E08 \(K\)-theory of schemes
14C25 Algebraic cycles
14F20 Étale and other Grothendieck topologies and (co)homologies
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
19E20 Relations of \(K\)-theory with cohomology theories
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