×

zbMATH — the first resource for mathematics

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”.

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.