The Weil-étale topology for number rings. (English) Zbl 1278.14029

Lichtenbaum, Stephen

Ann. Math. (2) 170, No. 2, 657-683 (2009).

Summary: There should be a Grothendieck topology for an arithmetic scheme \(X\) such that the Euler characteristic of the cohomology groups of the constant sheaf \(\mathbb Z\) with compact support at infinity gives, up to sign, the leading term of the zeta-function of \(X\) at \(s = 0\). We construct a topology (the Weil-étale topology) for the ring of integers in a number field whose cohomology groups \(H^i(\mathbb Z)\) determine such an Euler characteristic if we restrict to \(i \leq 3\).

MSC:

14F20	Étale and other Grothendieck topologies and (co)homologies
11R34	Galois cohomology
14G50	Applications to coding theory and cryptography of arithmetic geometry
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Keywords:

étale cohomology; zeta-function; Weil group

References:

[1]	E. Artin and J. Tate, Class Field Theory, New York: W. A. Benjamin, Inc., 1968. · Zbl 0176.33504
[2]	M. Artin, ”Grothendieck topologies,” Harvard University, mimeographed notes , 1962. · Zbl 0208.48701
[3]	A. Borel and N. R. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Princeton, N.J.: Princeton Univ. Press, 1980. · Zbl 0443.22010
[4]	P. Deligne, ”Le déterminant de la cohomologie,” in Current Trends in Arithmetical Algebraic Geometry, Providence, RI: Amer. Math. Soc., 1987, pp. 93-177. · Zbl 0629.14008
[5]	M. Flach, ”Cohomology of topological groups with applications to the Weil group,” Compos. Math., vol. 144, iss. 3, pp. 633-656, 2008. · Zbl 1145.18006
[6]	T. Geisser, ”Weil-étale cohomology over finite fields,” Math. Ann., vol. 330, iss. 4, pp. 665-692, 2004. · Zbl 1069.14021
[7]	T. Geisser, ”Arithmetic cohomology over finite fields and special values of \(\zeta\)-functions,” Duke Math. J., vol. 133, iss. 1, pp. 27-57, 2006. · Zbl 1104.14011
[8]	R. Godement, Topologie Algébrique et Théorie des Faisceaux, Paris: Publ. Math. Univ. Strasbourg 13, Hermann, 1958. · Zbl 0080.16201
[9]	S. Lichtenbaum, ”The Weil-étale topology on schemes over finite fields,” Compos. Math., vol. 141, iss. 3, pp. 689-702, 2005. · Zbl 1073.14024
[10]	J. S. Milne, Étale Cohomology, Princeton, N.J.: Princeton Univ. Press, 1980. · Zbl 0433.14012
[11]	J. Milnor, ”Whitehead torsion,” Bull. Amer. Math. Soc., vol. 72, pp. 358-426, 1966. · Zbl 0147.23104
[12]	C. C. Moore, ”Extensions and low dimensional cohomology theory of locally compact groups, I, II,” Trans. Amer. Math. Soc., vol. 113, pp. 40-63, 64, 1964. · Zbl 0131.26902
[13]	C. C. Moore, ”Group extensions and cohomology for locally compact groups: III,” Trans. Amer. Math. Soc., vol. 221, iss. 1, pp. 1-33, 1976. · Zbl 0366.22005
[14]	P. S. Mostert, ”Local cross sections in locally compact groups,” Proc. Amer. Math. Soc., vol. 4, pp. 645-649, 1953. · Zbl 0103.01801
[15]	J. Neukirch, . A. Schmidt, and . K. Wingberg, Cohomology of Number Fields, New York: Springer-Verlag, 2000. · Zbl 0948.11001
[16]	C. S. Rajan, ”On the vanishing of the measurable Schur cohomology groups of Weil groups,” Compos. Math., vol. 140, iss. 1, pp. 84-98, 2004. · Zbl 1057.11051
[17]	J. Tate, ”Number theoretic background,” in Automorphic Forms, Representations and \(L\)-Functions, II, Providence, R.I.: Amer. Math. Soc., 1979, pp. 3-26. · Zbl 0422.12007
[18]	D. Wigner, ”Algebraic cohomology of topological groups,” Trans. Amer. Math. Soc., vol. 178, pp. 83-93, 1973. · Zbl 0264.22001