Abstract
We show that Bloch’s complex of relative zero-cycles can be used as a dualizing complex over perfect fields and number rings. This leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields, finite fields, local fields of mixed characteristic, and rings of integers in number rings, generalizing results which so far have only been known for smooth schemes or in low dimensions, and unifying the $p$-adic and $l$-adic theory. As an application, we generalize Rojtman’s theorem to normal, projective schemes.
-
[blochrojt]
S. Bloch, "Torsion algebraic cycles and a theorem of Roitman," Compositio Math., vol. 39, iss. 1, pp. 107-127, 1979.@article {blochrojt, MRKEY = {539002},
AUTHOR = {Bloch, S.},
TITLE = {Torsion algebraic cycles and a theorem of {R}oitman},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {39},
YEAR = {1979},
NUMBER = {1},
PAGES = {107--127},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {14C15},
MRNUMBER = {80k:14012},
MRREVIEWER = {G. Horrocks},
URL = {http://www.numdam.org/item?id=CM_1979__39_1_107_0},
ZBLNUMBER = {0463.14002},
} -
[bloch] S. Bloch, "Algebraic cycles and higher $K$-theory," Adv. in Math., vol. 61, iss. 3, pp. 267-304, 1986.
@article {bloch, MRKEY = {852815},
AUTHOR = {Bloch, Spencer},
TITLE = {Algebraic cycles and higher {$K$}-theory},
JOURNAL = {Adv. in Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {61},
YEAR = {1986},
NUMBER = {3},
PAGES = {267--304},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {18F25 (11G45 14C35 19D99 19E15)},
MRNUMBER = {88f:18010},
MRREVIEWER = {L. N. Vaserstein},
DOI = {10.1016/0001-8708(86)90081-2},
ZBLNUMBER = {0608.14004},
} -
[blocloc] S. Bloch, "The moving lemma for higher Chow groups," J. Algebraic Geom., vol. 3, iss. 3, pp. 537-568, 1994.
@article {blocloc, MRKEY = {1269719},
AUTHOR = {Bloch, S.},
TITLE = {The moving lemma for higher {C}how groups},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {3},
YEAR = {1994},
NUMBER = {3},
PAGES = {537--568},
ISSN = {1056-3911},
MRCLASS = {14C15 (14C25 14C35)},
MRNUMBER = {96c:14007},
MRREVIEWER = {Claudio Pedrini},
ZBLNUMBER = {0830.14003},
} -
[blochkato] S. Bloch and K. Kato, "$p$-adic étale cohomology," Inst. Hautes Études Sci. Publ. Math., iss. 63, pp. 107-152, 1986.
@article {blochkato, MRKEY = {849653},
AUTHOR = {Bloch, Spencer and Kato, Kazuya},
TITLE = {{$p$}-adic étale cohomology},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {63},
YEAR = {1986},
PAGES = {107--152},
ISSN = {0073-8301},
CODEN = {PMIHA6},
MRCLASS = {14F30 (11S25 11S31 18F25 19F27)},
MRNUMBER = {87k:14018},
MRREVIEWER = {Gerd Faltings},
URL = {http://www.numdam.org/item?id=PMIHES_1986__63__107_0},
ZBLNUMBER = {0613.14017},
} -
[colliet] J. -L. Colliot-Thélène, "On the reciprocity sequence in the higher class field theory of function fields," in Algebraic $K$-Theory and Algebraic Topology, Dordrecht: Kluwer Acad. Publ., 1993, vol. 407, pp. 35-55.
@incollection {colliet, MRKEY = {1367291},
AUTHOR = {Colliot-Th{é}l{è}ne, J.-L.},
TITLE = {On the reciprocity sequence in the higher class field theory of function fields},
BOOKTITLE = {Algebraic {$K$}-Theory and Algebraic Topology},
VENUE={{L}ake {L}ouise, {AB},
1991},
SERIES = {NATO Adv. Sci. Inst. Ser. {\rm C} Math. Phys. Sci.},
VOLUME = {407},
PAGES = {35--55},
PUBLISHER = {Kluwer Acad. Publ.},
ADDRESS = {Dordrecht},
YEAR = {1993},
MRCLASS = {14F20 (11G45)},
MRNUMBER = {97k:14019},
ZBLNUMBER = {0885.19002},
} -
[delignehodgeIII] P. Deligne, "Théorie de Hodge. III," Inst. Hautes Études Sci. Publ. Math., iss. 44, pp. 5-77, 1974.
@article {delignehodgeIII, MRKEY = {0498552},
AUTHOR = {Deligne, Pierre},
TITLE = {Théorie de {H}odge. {III}},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {44},
YEAR = {1974},
PAGES = {5--77},
ISSN = {0073-8301},
MRCLASS = {14C30 (14F15)},
MRNUMBER = {58 \#16653b},
MRREVIEWER = {J. H. M. Steenbrink},
URL = {http://www.numdam.org/item?id=PMIHES_1971__40__5_0},
ZBLNUMBER = {0237.14003},
} -
[deninger2] C. Deninger, "Duality in the étale cohomology of one-dimensional proper schemes and generalizations," Math. Ann., vol. 277, iss. 3, pp. 529-541, 1987.
@article {deninger2, MRKEY = {891590},
AUTHOR = {Deninger, Christopher},
TITLE = {Duality in the étale cohomology of one-dimensional proper schemes and generalizations},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {277},
YEAR = {1987},
NUMBER = {3},
PAGES = {529--541},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {14F20 (11G45)},
MRNUMBER = {90a:14023},
MRREVIEWER = {Autorreferat},
DOI = {10.1007/BF01458330},
ZBLNUMBER = {0607.14011},
} -
[deningerwingberg] C. Deninger and K. Wingberg, "Artin-Verdier duality for $n$-dimensional local fields involving higher algebraic $K$-sheaves," J. Pure Appl. Algebra, vol. 43, iss. 3, pp. 243-255, 1986.
@article {deningerwingberg, MRKEY = {868985},
AUTHOR = {Deninger, Christopher and Wingberg, Kay},
TITLE = {Artin-{V}erdier duality for {$n$}-dimensional local fields involving higher algebraic {$K$}-sheaves},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {43},
YEAR = {1986},
NUMBER = {3},
PAGES = {243--255},
ISSN = {0022-4049},
CODEN = {JPAAA2},
MRCLASS = {11G35 (14F20)},
MRNUMBER = {87m:11054},
MRREVIEWER = {Gerd Faltings},
DOI = {10.1016/0022-4049(86)90066-6},
ZBLNUMBER = {0608.12016},
} -
[ichdede] T. Geisser, "Motivic cohomology over Dedekind rings," Math. Z., vol. 248, iss. 4, pp. 773-794, 2004.
@article {ichdede, MRKEY = {2103541},
AUTHOR = {Geisser, Thomas},
TITLE = {Motivic cohomology over {D}edekind rings},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {248},
YEAR = {2004},
NUMBER = {4},
PAGES = {773--794},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {14F42 (14G40)},
MRNUMBER = {2006c:14026},
MRREVIEWER = {Vladimir I. Guletski{\u\i}},
DOI = {10.1007/s00209-004-0680-x},
ZBLNUMBER = {1062.14025},
} -
[handbook] T. Geisser, "Motivic cohomology, $K$-theory and topological cyclic homology," in Handbook of $K$-Theory. Vol. 1, 2, New York: Springer-Verlag, 2005, pp. 193-234.
@incollection {handbook, MRKEY = {2181824},
AUTHOR = {Geisser, Thomas},
TITLE = {Motivic cohomology, {$K$}-theory and topological cyclic homology},
BOOKTITLE = {Handbook of {$K$}-Theory. {V}ol. 1, 2},
PAGES = {193--234},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2005},
MRCLASS = {14F42 (14C35 19D55)},
MRNUMBER = {2006j:14025},
MRREVIEWER = {Jean-Louis Cathelineau},
DOI = {10.1007/3-540-27855-9_6},
ZBLNUMBER = {1113.14017},
} -
[ichpicard] T. Geisser, "The affine part of the Picard scheme," Compos. Math., vol. 145, iss. 2, pp. 415-422, 2009.
@article {ichpicard, MRKEY = {2501424},
AUTHOR = {Geisser, Thomas},
TITLE = {The affine part of the {P}icard scheme},
JOURNAL = {Compos. Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {145},
YEAR = {2009},
NUMBER = {2},
PAGES = {415--422},
ISSN = {0010-437X},
MRCLASS = {14K30},
MRNUMBER = {2501424},
DOI = {10.1112/S0010437X08003710},
ZBLNUMBER = {1163.14025},
} -
[marcI] T. Geisser and M. Levine, "The $K$-theory of fields in characteristic $p$," Invent. Math., vol. 139, iss. 3, pp. 459-493, 2000.
@article {marcI, MRKEY = {1738056},
AUTHOR = {Geisser, Thomas and Levine, Marc},
TITLE = {The {$K$}-theory of fields in characteristic {$p$}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {139},
YEAR = {2000},
NUMBER = {3},
PAGES = {459--493},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {19D50 (14C15 19D45 19E08)},
MRNUMBER = {2001f:19002},
MRREVIEWER = {Michel Gros},
DOI = {10.1007/s002220050014},
ZBLNUMBER = {0957.19003},
} -
[bkbl] T. Geisser and M. Levine, "The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky," J. Reine Angew. Math., vol. 530, pp. 55-103, 2001.
@article {bkbl, MRKEY = {1807268},
AUTHOR = {Geisser, Thomas and Levine, Marc},
TITLE = {The {B}loch-{K}ato conjecture and a theorem of {S}uslin-{V}oevodsky},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {530},
YEAR = {2001},
PAGES = {55--103},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {14F42 (14C35 19E15)},
MRNUMBER = {2003a:14031},
MRREVIEWER = {R. T. Hoobler},
DOI = {10.1515/crll.2001.006},
ZBLNUMBER = {1023.14003},
} -
[grossuwa] M. Gros and N. Suwa, "La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique," Duke Math. J., vol. 57, iss. 2, pp. 615-628, 1988.
@article {grossuwa, MRKEY = {962522},
AUTHOR = {Gros, Michel and Suwa, Noriyuki},
TITLE = {La conjecture de {G}ersten pour les faisceaux de {H}odge-{W}itt logarithmique},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {57},
YEAR = {1988},
NUMBER = {2},
PAGES = {615--628},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {14C35 (14C15 14F30 19E15)},
MRNUMBER = {89h:14006b},
MRREVIEWER = {Wayne Raskind},
DOI = {10.1215/S0012-7094-88-05727-4},
ZBLNUMBER = {0715.14011},
} -
[fga] A. Grothendieck, Technique de descente et theoremes d’existence en geometrie algebrique. VI. Les schemas de Picard. Proprietes generales.
@misc{fga,
author={Grothendieck, A.},
TITLE={Technique de descente et theoremes d'existence en geometrie algebrique. VI. Les schemas de Picard. Proprietes generales},
NOTE={Seminaire Bourbaki, 1961/62, no. 236},
MRNUMBER={1611207},
ZBLNUMBER={0238.14015},
} -
[jannsen] U. Jannsen, "Hasse principles for higher-dimensional fields," Universität Regensburg, preprint 18/2004 , 2004.
@techreport{jannsen,
author = {Jannsen, Uwe},
TITLE={Hasse principles for higher-dimensional fields},
TYPE={preprint 18/2004},
INSTITUTION={Universität Regensburg},
YEAR={2004},
} -
[jannsensaito] U. Jannsen and S. Saito, "Kato homology of arithmetic schemes and higher class field theory over local fields," Doc. Math., iss. Extra Vol., pp. 479-538, 2003.
@article {jannsensaito, MRKEY = {2046606},
AUTHOR = {Jannsen, Uwe and Saito, Shuji},
TITLE = {Kato homology of arithmetic schemes and higher class field theory over local fields},
JOURNAL = {Doc. Math.},
FJOURNAL = {Documenta Mathematica},
YEAR = {2003},
NUMBER = {Extra Vol.},
PAGES = {479--538},
ISSN = {1431-0635},
MRCLASS = {11G45 (11G25 14F42 19D45)},
MRNUMBER = {2005c:11087},
MRREVIEWER = {Tam{á}s Szamuely},
ZBLNUMBER = {1092.14504},
} -
[JSS] U. Jannsen, S. Saito, and K. Sato, "Etale Duality for Constructible Sheaves on Arithmetic Schemes," Universität Regensburg, preprint 8/2004.
@techreport{JSS,
author={Jannsen, Uwe and Saito, Shuji and Sato, K.},
TITLE={Etale Duality for Constructible Sheaves on Arithmetic Schemes},
TYPE={preprint 8/2004},
INSTITUTION={Universität Regensburg},
} -
[katoduality] K. Kato, "Duality theories for the $p$-primary étale cohomology. I," in Algebraic and Topological Theories, Tokyo: Kinokuniya, 1986, pp. 127-148.
@incollection {katoduality, MRKEY = {1102256},
AUTHOR = {Kato, Kazuya},
TITLE = {Duality theories for the {$p$}-primary étale cohomology. {I}},
BOOKTITLE = {Algebraic and Topological Theories},
VENUE={{K}inosaki, 1984},
PAGES = {127--148},
PUBLISHER = {Kinokuniya},
ADDRESS = {Tokyo},
YEAR = {1986},
MRCLASS = {14F20},
MRNUMBER = {1102256},
ZBLNUMBER = {0800.14009},
} -
[kato] K. Kato, "A Hasse principle for two-dimensional global fields," J. Reine Angew. Math., vol. 366, pp. 142-183, 1986.
@article {kato, MRKEY = {833016},
AUTHOR = {Kato, Kazuya},
TITLE = {A {H}asse principle for two-dimensional global fields},
NOTE = {With an appendix by Jean-Louis Colliot-Th{é}l{è}ne},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {366},
YEAR = {1986},
PAGES = {142--183},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {11G35 (11E81 11R70)},
MRNUMBER = {88b:11036},
MRREVIEWER = {J. Browkin},
DOI = {10.1515/crll.1986.366.142},
ZBLNUMBER = {0576.12012},
} -
[katokuzumaki] K. Kato and T. Kuzumaki, "The dimension of fields and algebraic $K$-theory," J. Number Theory, vol. 24, iss. 2, pp. 229-244, 1986.
@article {katokuzumaki, MRKEY = {863657},
AUTHOR = {Kato, Kazuya and Kuzumaki, Takako},
TITLE = {The dimension of fields and algebraic {$K$}-theory},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {24},
YEAR = {1986},
NUMBER = {2},
PAGES = {229--244},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {12G10 (11E76 19D45)},
MRNUMBER = {88f:12008},
MRREVIEWER = {Adrian R. Wadsworth},
DOI = {10.1016/0022-314X(86)90105-8},
ZBLNUMBER = {0608.12029},
} -
[katzlang] N. M. Katz and S. Lang, "Finiteness theorems in geometric classfield theory," Enseign. Math., vol. 27, iss. 3-4, pp. 285-319 (1982), 1981.
@article {katzlang, MRKEY = {659153},
AUTHOR = {Katz, Nicholas M. and Lang, Serge},
TITLE = {Finiteness theorems in geometric classfield theory},
JOURNAL = {Enseign. Math.},
FJOURNAL = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
VOLUME = {27},
YEAR = {1981},
NUMBER = {3-4},
PAGES = {285--319 (1982)},
ISSN = {0013-8584},
CODEN = {ENMAAR},
MRCLASS = {14E20 (12A90 14K15)},
MRNUMBER = {83k:14012},
MRREVIEWER = {Ernst-Wilhelm Zink},
ZBLNUMBER = {0495.14011},
} -
[srin] A. Krishna and V. Srinivas, "Zero-cycles and $K$-theory on normal surfaces," Ann. of Math., vol. 156, iss. 1, pp. 155-195, 2002.
@article {srin, MRKEY = {1935844},
AUTHOR = {Krishna, Amalendu and Srinivas, V.},
TITLE = {Zero-cycles and {$K$}-theory on normal surfaces},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {156},
YEAR = {2002},
NUMBER = {1},
PAGES = {155--195},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {14C15 (14C25 14C35 19E08)},
MRNUMBER = {2003k:14005},
MRREVIEWER = {Charles Weibel},
DOI = {10.2307/3597187},
ZBLNUMBER = {1060.14015},
} -
[kleiman] S. L. Kleiman, "The Picard scheme," in Fundamental Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2005, vol. 123, pp. 235-321.
@incollection {kleiman, MRKEY = {2223410},
AUTHOR = {Kleiman, Steven L.},
TITLE = {The {P}icard scheme},
BOOKTITLE = {Fundamental Algebraic Geometry},
SERIES = {Math. Surveys Monogr.},
VOLUME = {123},
PAGES = {235--321},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2005},
MRCLASS = {14C22},
MRNUMBER = {2223410},
ZBLNUMBER={1085.14001},
} -
[levinerojtman] M. Levine, "Torsion zero-cycles on singular varieties," Amer. J. Math., vol. 107, iss. 3, pp. 737-757, 1985.
@article {levinerojtman, MRKEY = {789661},
AUTHOR = {Levine, Marc},
TITLE = {Torsion zero-cycles on singular varieties},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {107},
YEAR = {1985},
NUMBER = {3},
PAGES = {737--757},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {14C25 (14C35 18F25 19E15)},
MRNUMBER = {86h:14003},
MRREVIEWER = {K. R. Coombes},
DOI = {10.2307/2374375},
ZBLNUMBER = {0579.14007},
} -
[levinemoving] M. Levine, "Techniques of localization in the theory of algebraic cycles," J. Algebraic Geom., vol. 10, iss. 2, pp. 299-363, 2001.
@article {levinemoving, MRKEY = {1811558},
AUTHOR = {Levine, Marc},
TITLE = {Techniques of localization in the theory of algebraic cycles},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {10},
YEAR = {2001},
NUMBER = {2},
PAGES = {299--363},
ISSN = {1056-3911},
MRCLASS = {14C25 (14F42 19E15)},
MRNUMBER = {2001m:14014},
MRREVIEWER = {Tam{á}s Szamuely},
ZBLNUMBER = {1077.14509},
} -
[marcmc] M. Levine, "Motivic cohomology and K-theory of schemes. K-theory," , preprint archives 336.
@techreport{marcmc,
author={Levine, Marc},
TITLE = {Motivic cohomology and K-theory of schemes. K-theory},
TYPE={preprint archives 336},
URL={http://www.math.uiuc.edu/K-theory/},
} -
[levineweibel] M. Levine and C. Weibel, "Zero cycles and complete intersections on singular varieties," J. Reine Angew. Math., vol. 359, pp. 106-120, 1985.
@article {levineweibel, MRKEY = {794801},
AUTHOR = {Levine, Marc and Weibel, Chuck},
TITLE = {Zero cycles and complete intersections on singular varieties},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {359},
YEAR = {1985},
PAGES = {106--120},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {14C25 (14C35 19E15)},
MRNUMBER = {86k:14003},
MRREVIEWER = {K. R. Coombes},
ZBLNUMBER = {0555.14004},
} -
[mazur] B. Mazur, "Notes on étale cohomology of number fields," Ann. Sci. École Norm. Sup., vol. 6, pp. 521-552 (1974), 1973.
@article {mazur, MRKEY = {0344254},
AUTHOR = {Mazur, Barry},
TITLE = {Notes on étale cohomology of number fields},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {6},
YEAR = {1973},
PAGES = {521--552 (1974)},
ISSN = {0012-9593},
MRCLASS = {14F20 (12B25)},
MRNUMBER = {49 \#8993},
MRREVIEWER = {Gerhard Pfister},
URL = {http://www.numdam.org/item?id=ASENS_1973_4_6_4_521_0},
ZBLNUMBER = {0282.14004},
} -
[milneroitman] J. S. Milne, "Zero cycles on algebraic varieties in nonzero characteristic: Rojtman’s theorem," Compositio Math., vol. 47, iss. 3, pp. 271-287, 1982.
@article {milneroitman, MRKEY = {681610},
AUTHOR = {Milne, J. S.},
TITLE = {Zero cycles on algebraic varieties in nonzero characteristic: {R}ojtman's theorem},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {47},
YEAR = {1982},
NUMBER = {3},
PAGES = {271--287},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {14C25 (14C35)},
MRNUMBER = {85b:14011},
MRREVIEWER = {H. Gillet},
URL = {http://www.numdam.org/item?id=CM_1982__47_3_271_0},
ZBLNUMBER = {0506.14006},
} -
[milnebook] J. S. Milne, Étale Cohomology, Princeton, NJ: Princeton Univ. Press, 1980, vol. 33.
@book{milnebook,
author={Milne, J. S.},
TITLE={Étale Cohomology},
SERIES={Princeton Math. Series},
VOLUME={33},
PUBLISHER={Princeton Univ. Press},
ADDRESS={Princeton, NJ},
YEAR={1980},
MRNUMBER={0559531},
ZBLNUMBER={0433.14012},
} -
[adt] J. S. Milne, Arithmetic Duality Theorems, Boston, MA: Academic Press, 1986, vol. 1.
@book {adt, MRKEY = {881804},
AUTHOR = {Milne, J. S.},
TITLE = {Arithmetic Duality Theorems},
SERIES = {Perspectives in Mathematics},
VOLUME = {1},
PUBLISHER = {Academic Press},
ADDRESS = {Boston, MA},
YEAR = {1986},
PAGES = {x+421},
ISBN = {0-12-498040-6},
MRCLASS = {14F20 (11G99 11R34 12G10)},
MRNUMBER = {88e:14028},
MRREVIEWER = {Gerd Faltings},
ZBLNUMBER = {0613.14019},
} -
[milnevalues] J. S. Milne, "Values of zeta functions of varieties over finite fields," Amer. J. Math., vol. 108, iss. 2, pp. 297-360, 1986.
@article {milnevalues, MRKEY = {833360},
AUTHOR = {Milne, J. S.},
TITLE = {Values of zeta functions of varieties over finite fields},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {108},
YEAR = {1986},
NUMBER = {2},
PAGES = {297--360},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {14G10 (11G25 14F30 14G15)},
MRNUMBER = {87g:14019},
MRREVIEWER = {Thomas Zink},
DOI = {10.2307/2374676},
ZBLNUMBER = {0611.14020},
} -
[moser] T. Moser, "A duality theorem for étale $p$-torsion sheaves on complete varieties over a finite field," Compositio Math., vol. 117, iss. 2, pp. 123-152, 1999.
@article {moser, MRKEY = {1695861},
AUTHOR = {Moser, Thomas},
TITLE = {A duality theorem for étale {$p$}-torsion sheaves on complete varieties over a finite field},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {117},
YEAR = {1999},
NUMBER = {2},
PAGES = {123--152},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {14F30 (11G25 14F20)},
MRNUMBER = {2000f:14030},
MRREVIEWER = {Jean-Claude Douai},
DOI = {10.1023/A:1000892524712},
ZBLNUMBER={0954.14012},
} -
[nart] E. Nart, "The Bloch complex in codimension one and arithmetic duality," J. Number Theory, vol. 32, iss. 3, pp. 321-331, 1989.
@article {nart, MRKEY = {1006598},
AUTHOR = {Nart, Enric},
TITLE = {The {B}loch complex in codimension one and arithmetic duality},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {32},
YEAR = {1989},
NUMBER = {3},
PAGES = {321--331},
ISSN = {0022-314X},
CODEN = {JNUTA9},
MRCLASS = {11G45 (14C15 14F20 19F27)},
MRNUMBER = {90j:11057},
MRREVIEWER = {Takeshi Ooe},
DOI = {10.1016/0022-314X(89)90088-7},
ZBLNUMBER = {0728.14002},
} -
[roitman] A. A. Rojtman, "The torsion of the group of $0$-cycles modulo rational equivalence," Ann. of Math., vol. 111, iss. 3, pp. 553-569, 1980.
@article {roitman, MRKEY = {577137},
AUTHOR = {Rojtman, A. A.},
TITLE = {The torsion of the group of {$0$}-cycles modulo rational equivalence},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {111},
YEAR = {1980},
NUMBER = {3},
PAGES = {553--569},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {14C15 (14C25)},
MRNUMBER = {81g:14003},
MRREVIEWER = {D. Lieberman},
DOI = {10.2307/1971109},
ZBLNUMBER = {0504.14006},
} -
[roos] J. Roos, "Sur les foncteurs dérivés de $\underleftarrow\lim$. Applications," C. R. Acad. Sci. Paris, vol. 252, pp. 3702-3704, 1961.
@article {roos, MRKEY = {0132091},
AUTHOR = {Roos, Jan-Erik},
TITLE = {Sur les foncteurs dérivés de {$\underleftarrow\lim$}. {A}pplications},
JOURNAL = {C. R. Acad. Sci. Paris},
VOLUME = {252},
YEAR = {1961},
PAGES = {3702--3704},
MRCLASS = {18.20},
MRNUMBER = {24 \#A1938},
MRREVIEWER = {A. Dold},
ZBLNUMBER = {0102.02501},
} -
[shuji] S. Saito, "Torsion zero-cycles and étale homology of singular schemes," Duke Math. J., vol. 64, iss. 1, pp. 71-83, 1991.
@article {shuji, MRKEY = {1131393},
AUTHOR = {Saito, Shuji},
TITLE = {Torsion zero-cycles and étale homology of singular schemes},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {64},
YEAR = {1991},
NUMBER = {1},
PAGES = {71--83},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {14C25 (14F20)},
MRNUMBER = {92h:14004},
MRREVIEWER = {Luca Barbieri Viale},
DOI = {10.1215/S0012-7094-91-06403-3},
ZBLNUMBER = {0764.14011},
} -
[seidenberg] A. Seidenberg, "The hyperplane sections of normal varieties," Trans. Amer. Math. Soc., vol. 69, pp. 357-386, 1950.
@article {seidenberg, MRKEY = {0037548},
AUTHOR = {Seidenberg, A.},
TITLE = {The hyperplane sections of normal varieties},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {69},
YEAR = {1950},
PAGES = {357--386},
ISSN = {0002-9947},
MRCLASS = {14.0X},
MRNUMBER = {12,279a},
MRREVIEWER = {P. Samuel},
DOI = {10.2307/1990364},
ZBLNUMBER = {0040.23501},
} -
[sgafo] "Théorie des topos et cohomologie étale des schémas. Tome 3," in Séminair de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Artin, M., Grothendieck, A., and Verdier, J. L., Eds., New York: Springer-Verlag, 1973, vol. 305.
@incollection{sgafo, EDITOR={Artin, M. and Grothendieck, A. and Verdier, J. L.},
TITLE={Théorie des topos et cohomologie étale des schémas. Tome 3},
BOOKTITLE={Séminair de Géométrie Algébrique du Bois-Marie 1963--1964 {\rm (SGA 4)}},
NOTE={avec la collaboration de P. Deligne et B. Saint-Donat},
SERIES={Lecture Notes in Math.},
VOLUME={305},
PUBLISHER={Springer-Verlag},
ADDRESS={New York},
YEAR={1973},
MRNUMBER={0354654},
ZBLNUMBER={0245.00002},
} -
[sgafi] "Cohomologie $l$-adique et fonctions $L$. (French)," in Séminair de Géométrie Algébrique du Bois-Marie 1965–1965 (SGA 5), Illusie, L., Ed., New York: Springer-Verlag, 1977, vol. 589.
@incollection{sgafi, EDITOR={Illusie, L.},
TITLE={Cohomologie $l$-adique et fonctions $L$. (French)},
BOOKTITLE={Séminair de Géométrie Algébrique du Bois-Marie 1965--1965 {\rm (SGA 5)}},
SERIES={Lecture Notes in Math.},
VOLUME={589},
PUBLISHER={Springer-Verlag},
ADDRESS={New York},
YEAR={1977},
MRNUMBER={0491704},
ZBLNUMBER={0345.00011},
} -
[serre] J. P. Serre, Morphismes universels et variétès d’Albanese.
@misc{serre,
author={Serre, J. P.},
TITLE={Morphismes universels et variétès d'Albanese},
NOTE={Séminaire Chevalley {\bf 3} (1958--1959), exposé 11 (1960)},
} -
[spaltenstein] N. Spaltenstein, "Resolutions of unbounded complexes," Compositio Math., vol. 65, iss. 2, pp. 121-154, 1988.
@article {spaltenstein, MRKEY = {932640},
AUTHOR = {Spaltenstein, N.},
TITLE = {Resolutions of unbounded complexes},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {65},
YEAR = {1988},
NUMBER = {2},
PAGES = {121--154},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {18E25 (32C35)},
MRNUMBER = {89m:18013},
MRREVIEWER = {Michael M. Kapranov},
URL = {http://www.numdam.org/item?id=CM_1988__65_2_121_0},
ZBLNUMBER = {0636.18006},
} -
[spiess] M. Spieß, "Artin-Verdier duality for arithmetic surfaces," Math. Ann., vol. 305, iss. 4, pp. 705-792, 1996.
@article {spiess, MRKEY = {1399713},
AUTHOR = {Spieß,
Michael},
TITLE = {Artin-{V}erdier duality for arithmetic surfaces},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {305},
YEAR = {1996},
NUMBER = {4},
PAGES = {705--792},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {14G40 (11G45 14F20)},
MRNUMBER = {97i:14019},
MRREVIEWER = {J{ö}rg Jahnel},
DOI = {10.1007/BF01444246},
ZBLNUMBER = {0887.14008},
} -
[suslinetale] A. A. Suslin, "Higher Chow groups and etale cohomology," in Cycles, Transfers, and Motivic Homology Theories, Princeton, NJ: Princeton Univ. Press, 2000, vol. 143, pp. 239-254.
@incollection {suslinetale, MRKEY = {1764203},
AUTHOR = {Suslin, Andrei A.},
TITLE = {Higher {C}how groups and etale cohomology},
BOOKTITLE = {Cycles, Transfers, and Motivic Homology Theories},
SERIES = {Ann. of Math. Stud.},
VOLUME = {143},
PAGES = {239--254},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {2000},
MRCLASS = {14C15 (14F20)},
MRNUMBER = {1764203},
ZBLNUMBER = {1019.14001},
} -
[suwa] N. Suwa, "A note on Gersten’s conjecture for logarithmic Hodge-Witt sheaves," $K$-Theory, vol. 9, iss. 3, pp. 245-271, 1995.
@article {suwa, MRKEY = {1344141},
AUTHOR = {Suwa, Noriyuki},
TITLE = {A note on {G}ersten's conjecture for logarithmic {H}odge-{W}itt sheaves},
JOURNAL = {$K$-Theory},
FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of $K$-Theory in the Mathematical Sciences},
VOLUME = {9},
YEAR = {1995},
NUMBER = {3},
PAGES = {245--271},
ISSN = {0920-3036},
CODEN = {KTHEEO},
MRCLASS = {14F30 (14C35 19E15)},
MRNUMBER = {97d:14030},
MRREVIEWER = {Yuichiro Takeda},
DOI = {10.1007/BF00961667},
ZBLNUMBER = {0838.14014},
} -
[thomason] R. W. Thomason, "Algebraic $K$-theory and étale cohomology," Ann. Sci. École Norm. Sup., vol. 18, iss. 3, pp. 437-552, 1985.
@article {thomason, MRKEY = {826102},
AUTHOR = {Thomason, R. W.},
TITLE = {Algebraic {$K$}-theory and étale cohomology},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {18},
YEAR = {1985},
NUMBER = {3},
PAGES = {437--552},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {14F15 (11G99 14F12 18F25 19E20)},
MRNUMBER = {87k:14016},
MRREVIEWER = {H. Gillet},
URL = {http://www.numdam.org/item?id=ASENS_1985_4_18_3_437_0},
ZBLNUMBER = {0596.14012},
} -
[voevodsky] V. Voevodsky, "Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic," Int. Math. Res. Not., vol. 2002, iss. 7, pp. 351-355, 2002.
@article {voevodsky, MRKEY = {1883180},
AUTHOR = {Voevodsky, Vladimir},
TITLE = {Motivic cohomology groups are isomorphic to higher {C}how groups in any characteristic},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2002},
NUMBER = {7},
PAGES = {351--355},
ISSN = {1073-7928},
MRCLASS = {14F42 (14C25 19E15)},
MRNUMBER = {2003c:14021},
MRREVIEWER = {Thomas Geisser},
DOI = {10.1155/S107379280210403X},
VOLUME = {2002},
ZBLNUMBER = {1057.14026},
}
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