The Weil-étale topology on schemes over finite fields. (English) Zbl 1073.14024

Lichtenbaum, S.

Compos. Math. 141, No. 3, 689-702 (2005).

The paper studies the Weil-étale topology which is to the usual étale topology what is the cyclic group \(\mathbb{Z}\) to its profinite completion. Namely for a scheme over a finite field \(k\) a Weil-étale sheaf on \(X\) is an étale sheaf on the base-extension to the algebraic closure \(k\), together with an isomorphism to its Frobenius-pullback. (Note that the total Frobenius is the product of the geometric and the arithmetic Frobenius, and it induces a natural equivalence on sheaves.) The properties of the corresponding cohomology are studied, also for non-torsion coefficients like the integers \(\mathbb{Z}\). This relies heavily on previous work of J. Milne [“Étale cohomology”, Princeton Univ. Press (1980; Zbl 0433.14012); Am. J. Math. 108, 297–360 (1986; Zbl 0611.14020)]. Finally, the author states, and proves in some cases, a conjecture relating Euler-characteristics and zeta-values at \(s= 0\).

Reviewer: Gerd Faltings (Bonn)

MSC:

14F20	Étale and other Grothendieck topologies and (co)homologies
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Keywords:

étale topology; Weil-group

Citations:

Zbl 0433.14012; Zbl 0611.14020