Weil-etale motivic cohomology

We study Weil-etale cohomology, introduced by Lichtenbaum for varieties over finite fields. In the first half of the paper we give an explicit description of the base change from Weil-etale cohomology to etale cohomology. As a consequence, we get a long exact sequence relating Weil-etale cohomology to etale cohomology, show that for finite coefficients the cohomology theories agree, and with rational coefficients a Weil-etale cohomology group is the direct sum of two etale cohomology groups. In the second half of the paper we restrict ourselves to Weil-etale cohomology of the motivic complex. We show that for smooth projective varieties over finite fields, finite generation of Weil-etale cohomology is equivalent to Weil-etale cohomology being an integral model of l-adic cohomology, and also equivalent to the conjunction of Tate's conjecture and (rational) equality of rational and numerical equivalence. We give several examples where these conjectures hold, and express special values of zeta functions in terms of Weil-etale cohomology.

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

[2]  B. Kahn.  The Geisser-Levine method revisited and algebraic cycles over a finite field , 2002 .

[3]  B. Kahn.  Equivalence rationnelle, equivalence numerique et produits de courbes elliptiques sur un corps fini , 2002, math/0205158.

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

[5]  S. Lichtenbaum.  THE WEIL- ETALE TOPOLOGY , 2001 .

[6]  Thomas H. Geisser.  Applications of de Jong’s Theorem on Alterations , 2000 .

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

[8]  Thomas H. Geisser.  Tate's Conjecture, Algebraic Cycles and Rational K-Theory in Characteristic p , 1998 .

[9]  B. Kahn.  A SHEAF-THEORETIC REFORMULATION OF THE TATE CONJECTURE , 1998, math/9801017.

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

[11]  Uwe Jannsen.  Continuous étale cohomology , 1988 .

[12]  J. Milne.  Motivic cohomology and values of zeta functions , 1988 .

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

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

[15]  C. Soulé.  Groupes de Chow etK-théorie de variétés sur un corps fini , 1984 .

[16]  S. Lichtenbaum.  Values of zeta-functions at non-negative integers , 1984 .

[17]  M. Rapoport,et al.  Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik , 1982 .

[18]  J. Tate.  Endomorphisms of abelian varieties over finite fields , 1966 .

[19]  J. Tate,et al.  Algebraic cycles and poles of zeta functions , 1965 .