×

zbMATH — the first resource for mathematics

Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique. (Chern classes and cycle classes in logarithmic Hodge-Witt cohomology). (French) Zbl 0615.14011
The author constructs cycle classes in p-adic cohomology theories; crystalline cohomology and Hodge-Witt cohomology. The technical difficulties usually encountered when trying to do so are resolved by introducing the so called logarithmic cohomology, the fixed points under the Frobenius map of the Hodge-Witt cohomology. It turns out that for the latter theory one can prove purity results strong enough to reduce, in the usual way, the construction of cycle classes to the case of smooth cycles, a case which follows from duality. It should be noted that the needed purity results are not true for crystalline nor for Hodge-Witt cohomology. The article further computes the Hodge-Witt cohomology for a projective bundle and a smooth blowing up in terms of the cohomology of the base variety thereby obtaining Chern classes and proves the usual formulas for the cycle classes up to and including the ”projection formula” and Riemann-Roch ”without denominators” for a closed embedding.
Note that in particular the article provides the missing results enabling us to declare crystalline cohomology (tensored with \({\mathbb{Q}})\) a ”Weil cohomology theory” [cf. S. L. Kleiman in: Dix Exposés Cohomologies Schémas, Adv. Stud. Math. 3, 359-386 (1986; Zbl 0198.259)].
Reviewer: T.Ekedahl

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
57R20 Characteristic classes and numbers in differential topology
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] A. BEAUVILLE . - Traces et résidus en géométrie algébrique . Thèse de troisième cycle, Orsay.
[2] P. BERTHELOT . - Cohomologie cristalline des schémas de caractéristique p > 0 (Lecture Notes in Math., n^\circ 407, Springer-Verlag 1974 ). MR 52 #5676 | Zbl 0298.14012 · Zbl 0298.14012
[3] P. BERTHELOT . - Dualité plate, dans “Surfaces algébriques” . Séminaire de Géométrie Algébrique d’Orsay, 1976 - 1978 (Lecture Notes in Math., n^\circ 868, Springer-Verlag 1981 ).
[4] P. BERTHELOT , L. ILLUSIE . - Classes de Chern en cohomologie cristalline (Notes aux C.R.A.S.P. du 22 et du 29 Juin 1970 , Série A, t. 270). MR 42 #4555 | Zbl 0198.26201 · Zbl 0198.26201
[5] S. BLOCH , K. KATO . - p-adic etale cohomology . Preprint. · Zbl 0613.14017
[6] S. BLOCH , A. OGUS . - Gersten’s conjecture and the homology of schemes (Annales de l’E.N.S., 4e série, t. 7, 1974 , p. 181 à 202). Numdam | MR 54 #318 | Zbl 0307.14008 · Zbl 0307.14008 · numdam:ASENS_1974_4_7_2_181_0 · eudml:81935
[7] J.-L. COLLIOT-THELENE , J.-J. SANSUC , C. SOULE . - Torsion dans le groupe de Chow de codimension deux (Duke Math. Journ. 50, 1983 , 763-801). Article | MR 85d:14010 | Zbl 0574.14004 · Zbl 0574.14004 · doi:10.1215/S0012-7094-83-05038-X · minidml.mathdoc.fr
[8] P. DELIGNE . - Théorème de Lefschetz et critères de dégénérescence des suites spectrales (Publications mathématiques de l’I.H.E.S. 35). Numdam | Zbl 0159.22501 · Zbl 0159.22501 · doi:10.1007/BF02698925 · numdam:PMIHES_1968__35__107_0 · eudml:103884
[9] T. EKEDAHL . - On the multiplicative properties of the De Rham-Witt complex . A paraître dans Archiv für Mathematik. Zbl 0575.14017 · Zbl 0575.14017 · doi:10.1007/BF02384419
[10] T. EKEDAHL . - Travail en préparation.
[11] F. EL ZEIN . - Complexe dualisant et applications à la classe fondamentale d’un cycle (BSMF, Mémoire n^\circ 58, 1978 ). Numdam | MR 80h:14009 | Zbl 0388.14002 · Zbl 0388.14002 · numdam:MSMF_1978__58__1_0 · eudml:94780
[12] H. GILLET . - Riemann-Roch theorems for higher algebraic K-theory (Advances in Math. 40, 203-289 ( 1981 ). MR 83m:14013 | Zbl 0478.14010 · Zbl 0478.14010 · doi:10.1016/S0001-8708(81)80006-0
[13] A. GROTHENDIECK . - La théorie des classes de Chern (BSMF 86, 1958 , 137-154). Numdam | MR 22 #6818 | Zbl 0091.33201 · Zbl 0091.33201 · numdam:BSMF_1958__86__137_0 · eudml:86933
[14] A. GROTHENDIECK . - Classes de Chern et représentations linéaires des groupes discrets dans “Dix exposés sur la cohomologie des schémas (North-Holland, Masson). Zbl 0198.26101 · Zbl 0198.26101
[15] O. GABBER . - Lettre à Bloch du 11 Janvier 1982 .
[16] O. GABBER . - Gersten’s Conjecture for some complexes of vanishing cycles . Preprint. · Zbl 0827.19002 · doi:10.1007/BF02568202 · eudml:156026
[17] R. HARTSHORNE . - Residues and Duality (Lecture Notes in Math., n^\circ 20, Springer-Verlag 1966 ). MR 36 #5145 | Zbl 0212.26101 · Zbl 0212.26101 · doi:10.1007/BFb0080482 · eudml:203789
[18] R. HARTSHORNE . - On the De Rham cohomology of algebraic varieties (Publications mathématiques de l’I.H.E.S. 45, 1976 (5-99). Numdam | MR 55 #5633 | Zbl 0326.14004 · Zbl 0326.14004 · doi:10.1007/BF02684298 · numdam:PMIHES_1975__45__5_0 · eudml:103941
[19] L. ILLUSIE . - Complexe cotangent et déformations (Lecture Notes in Math., n^\circ 239 et 283, Springer-Verlag 1971 et 1972 ). MR 58 #10886a | Zbl 0224.13014 · Zbl 0224.13014 · doi:10.1007/BFb0059052
[20] L. ILLUSIE . - Complexe de De Rham-Witt (Astérisque 63, 1979 , 83-112). MR 81j:14012 | Zbl 0446.14008 · Zbl 0446.14008
[21] L. ILLUSIE . - Complexe de De Rham-Witt et cohomologie cristalline (Annales de l’E.N.S., 4e série, t. 12, 1979 , 501-661). Numdam | MR 82d:14013 | Zbl 0436.14007 · Zbl 0436.14007 · numdam:ASENS_1979_4_12_4_501_0 · eudml:82043
[22] L. ILLUSIE . - Finiteness, duality, and Künneth theorems in the cohomology of the De Rham-Witt complex . Zbl 0538.14013 · Zbl 0538.14013
[23] L. ILLUSIE , M. RAYNAUD . - Les suites spectrales associées au complexe de De Rham-Witt (Publ. math. de l’IHES n^\circ 57 p. 73-212, 1983 ). Numdam | MR 85e:14026 | Zbl 0538.14012 · Zbl 0538.14012 · doi:10.1007/BF02698774 · numdam:PMIHES_1983__57__73_0 · eudml:103990
[24] J.-P. JOUANOLOU . - Riemann-Roch sans dénominateurs (Inventionnes Math. 11, 15-26 ( 1970 ). MR 48 #11115 | Zbl 0199.55901 · Zbl 0199.55901 · doi:10.1007/BF01389802 · eudml:142041
[25] K. KATO , S. SAITO . - Unramified class field theory of arithmetical surfaces (Annals of Math. 118 ( 1983 ), 241-275). MR 86c:14006 | Zbl 0562.14011 · Zbl 0562.14011 · doi:10.2307/2007029
[26] N. KATZ . - Crystalline cohomology, Dieudonné Modules, and Jacobi Sums (Automorphic forms, Tata Institute, Bombay, 1979 , 165-246). MR 83a:14022 | Zbl 0502.14007 · Zbl 0502.14007
[27] B. MAZUR , L. ROBERTS . - Local Euler characteristics (Invent. Math. 9 ( 1970 ), 201-234). MR 41 #3490 | Zbl 0191.19202 · Zbl 0191.19202 · doi:10.1007/BF01404325 · eudml:142007
[28] J.S. MILNE . - Duality in the flat cohomology of a surface (Annales de l’E.N.S. 4e série, t. 9 ( 1976 ), 171-202). Numdam | MR 57 #325 | Zbl 0334.14010 · Zbl 0334.14010 · numdam:ASENS_1976_4_9_2_171_0 · eudml:81978
[29] J.S. MILNE . - Zero cycles on algebraic varieties in non zero characteristic : Rotjman’s theorem (Compositio Mathématica, vol. 47 ( 1982 ), 271-287). Numdam | MR 85b:14011 | Zbl 0506.14006 · Zbl 0506.14006 · numdam:CM_1982__47_3_271_0 · eudml:89575
[30] J.S. MILNE . - Values of zeta functions of varieties over finite fields . Preprint. · Zbl 0611.14020 · doi:10.2307/2374676
[31] N.1. NYGAARD . - Slopes of powers of Frobenius on crystalline cohomology (Annales de l’E.N.S. 4e série, t. 14 ( 1981 ), 369-401). Numdam | MR 84d:14011 | Zbl 0519.14012 · Zbl 0519.14012 · numdam:ASENS_1981_4_14_4_369_0 · eudml:82080
[32] A. OGUS . - Hodge cycles and crystalline cohomology dans “Hodge cycles, Motives and Shimura varieties” (Lecture Notes in Math. 900, Springer-Verlag 1982 ). MR 84m:14046 | Zbl 0538.14010 · Zbl 0538.14010
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.