Duality via cycle complexes

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.

[1]  Thomas H. Geisser.  Motivic cohomology over Dedekind rings , 2004 .

[2]  Thomas H. Geisser.  Motivic Cohomology, K-Theory and Topological Cyclic Homology* , 2004 .

[3]  S. Saito,et al.  Kato homology of arithmetic schemes and higher class field theory over local fields. , 2003 .

[4]  V. Srinivas,et al.  Zero-cycles and K-theory on normal surfaces , 2002 .

[5]  Vladimir Voevodsky,et al.  Motivic cohomology groups are isomorphic to higher chow groups in any characteristic , 2002 .

[6]  Thomas H. Geisser,et al.  The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky , 2001 .

[7]  Thomas H. Geisser,et al.  The K-theory of fields in characteristic p , 2000 .

[8]  Vladimir Voevodsky,et al.  Cycles, Transfers And Motivic Homology Theories , 2000 .

[9]  S. Lang,et al.  Finiteness Theorems in Geometric Classfield Theory , 2000 .

[10]  T. Moser.  A Duality Theorem for Étale p-Torsion Sheaves on Complete Varieties over a Finite Field , 1999, Compositio Mathematica.

[11]  A. Suslin.  Higher Chow groups and etale cohomology , 1999 .

[12]  M. Levine.  TECHNIQUES OF LOCALIZATION IN THE THEORY OF ALGEBRAIC CYCLES , 1999 .

[13]  Michael Spiefi.  Artin-Verdier duality for arithmetic surfaces , 1996 .

[14]  Noriyuki Suwa.  A note on Gersten's conjecture for logarithmic Hodge-Witt sheaves , 1995 .

[15]  J. Colliot-Thélène.  On the Reciprocity Sequence in the Higher Class Field Theory of Function Fields , 1993 .

[16]  S. Saito.  Torsion zero-cycles and ètale homology of singular schemes , 1991 .

[17]  E. Nart.  The Bloch Complex in Codimension One and Arithmetic Duality , 1989 .

[18]  M. Gros,et al.  La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique , 1988 .

[19]  N. Spaltenstein.  Resolutions of unbounded complexes , 1988 .

[20]  C. Deninger.  Duality in the Étale cohomology of one-dimensional proper schemes and generalizations , 1987 .

[21]  James S. Milne,et al.  Arithmetic Duality Theorems , 1987 .

[22]  S. Bloch,et al.  p-Adic etale cohomology , 1986 .

[23]  C. Deninger,et al.  Artin-Verdier duality for n-dimensional local fields involving higher algebraic K-sheaves , 1986 .

[24]  Kazuya Kato,et al.  The dimension of fields and algebraic K-theory , 1986 .

[25]  Spencer Bloch,et al.  Algebraic cycles and higher K-theory , 1986 .

[26]  J. Milne.  VALUES OF ZETA FUNCTIONS OF VARIETIES OVER FINITE FIELDS , 1986 .

[27]  Kazuya Kato.  A Hasse principle for two dimensional global fields. , 1986 .

[28]  M. Levine.  Torsion Zero-Cycles on Singular Varieties , 1985 .

[29]  R. Thomason.  Algebraic $K$-theory and etale cohomology , 1985 .

[30]  C. Weibel,et al.  Zero cycles and complete intersections on singular varieties. , 1985 .

[31]  J. Milne.  Zero cycles on algebraic varieties in nonzero characteristic : Rojtman's theorem , 1982 .

[32]  A. A. Rojtman.  The Torsion of the Group of 0-Cycles Modulo Rational Equivalence , 1980 .

[33]  S. Bloch.  Torsion algebraic cycles and a theorem of Roitman , 1979 .

[34]  B. Mazur.  Notes on étale cohomology of number fields , 1973 .

[35]  P. Deligne,et al.  Théorie de Hodge, II , 1971 .

[36]  P. Deligne.  Theorie de Hodge I , 1970 .

[37]  Alexander Grothendieck,et al.  Technique de descente et théorèmes d'existence en géométrie algébrique. VI. Les schémas de Picard : propriétés générales , 1962 .

[38]  Jean-Pierre Serre.  Morphismes universels et variété d'Albanese , 1959 .

[39]  A. Seidenberg.  THE HYPERPLANE SECTIONS OF NORMAL VARIETIES , 1950 .