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

Gabber, Ofer

Manuscr. Math. 85, No. 3-4, 323-343 (1994).

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”.

Reviewer: S.Xambó-Descamps (Barcelona)

MSC:

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

Keywords:

sheaves; cohomology; \(p\)-adic vanishing cycles; \(K\)-theory; schemes; Gersten conjecture

Citations:

Zbl 0307.14008; Zbl 0289.18011; Zbl 0292.18004

References:

[1]	S. Bloch and A. Ogus, Gersten’s Conjecture and the Homology of Schemes,Ann. Sci. Ecole Norm. Sup. 7 (1974), 181–202 · Zbl 0307.14008
[2]	O. Gabber, An injectivity property for étale cohomology,Compositio Math. 86 (1993), 1–14 · Zbl 0828.14011
[3]	D. Quillen, Higher AlgebraicK-theory I, inLecture Notes in Math. 341 (1973), Springer-Verlag
[4]	A. Altman and S. Kleiman, Introduction to Grothendieck’s duality theory,Lecture Notes in Math. 146 (1970), Springer-Verlag · Zbl 0215.37201
[5]	M. Artin, A. Grothendieck, J.L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4),Lecture Notes in Math. 269, 270, 305 (1972–1973), Springer-Verlag
[6]	R. Hartshorne, Residues and Duality,Lecture Notes in Math. 20 (1966), Springer-Verlag
[7]	J.-L. Colliot-Thélène, R. Hoobler and B. Kahn, Equivariant refinements of Gersten conjecture
[8]	D. Grayson, Universal exactness in algebraicK-theory,J. Pure Appl. Algebra 36 (1985), 139–141 · Zbl 0558.18007
[9]	M. Gros and S. Suwa, La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmiques,Duke Math. Journal 57 (1988), 615–628 · Zbl 0715.14011
[10]	M. Ojanguren, Quadratic forms over regular rings,Journal of the Indian Math. Soc. 44 (1980), 109–116 · Zbl 0621.10017