Arithmetic cohomology over finite fields and special values of $\zeta$-functions

We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))$ for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.

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

[2]  Thomas H. Geisser.  Weil-étale cohomology over finite fields , 2004, math/0404425.

[3]  B. Kahn.  Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini , 2003 .

[4]  B. Kahn.  Some finiteness results for étale cohomology , 2003 .

[5]  C. Weibel.  The negative K-theory of normal surfaces , 2001 .

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

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

[8]  A. Suslin,et al.  Bloch-Kato Conjecture and Motivic Cohomology with Finite Coefficients , 2000 .

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

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

[11]  A. D. de Jong.  Smoothness , semi-stability and alterations , 1996 .

[12]  A. Suslin,et al.  Singular homology of abstract algebraic varieties , 1996 .

[13]  U. Jannsen.  Mixed Motives and Algebraic K-Theory , 1990 .

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

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

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

[17]  M. Gros.  Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique , 1985 .

[18]  Séminaire Bourbaki,et al.  Dix exposés sur la cohomologie des schémas , 1968 .