On the Weil-étale cohomology of the ring of S-integers

2011/12/01 by Yi-Chih Chiu, Chiu, Yi-Chih
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1112.0092

openalex publication_date 2011/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we first briefly introduce the history of the Weil-étale cohomology theory of arithmetic schemes and review some important results established by Lichtenbaum, Flach and Morin. Next we generalize the Weil-etale cohomology to S-integers and compute the cohomology for constant sheaves ℤ or ℝ. We also define a Weil-étale cohomology with compact support H_c(Y_W, -) for Y=Spec O_F,S where F is a number field, and computed them. We verify that these cohomology groups satisfy the axioms state by Lichtenbaum. As an application, we derive a canonical representation of Tate sequence from RGamma_c(Y_W,ℤ). Motivated by this result, in the final part, we define an étale complex RGm, such that the complexes ℤ-dual of the complex \RG(U_et,R\Gm), ℤ)[2] is canonically quasi-isomorphic to τ^{≤ 3}\RG_c(U_W,ℤ) for arbitrary étale U over Spec O_F. This quasi-isomorphism provides a possible approach to define the Weil-etale cohomology for higher dimensional arithmetic schemes, as the Weil groups are not involved in the definition of R\Gm.

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