\'Etale motives

TL;DR

This paper develops a comprehensive theory of etale motives over noetherian schemes, establishing their properties, relations to Beilinson motives, and applications to purity, finiteness, and duality, with implications for $ ext{l}$-adic realizations.

Contribution

It introduces a new framework for etale motives with integral coefficients, extending rigidity theorems and connecting torsion motives with classical sheaves.

Findings

Equivalence of torsion etale motives with complexes of torsion sheaves

Validation of absolute purity, finiteness, and duality for etale motives

Construction of $ ext{l}$-adic realization via homotopy $ ext{l}$-completion

Abstract

We define a theory of etale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$-theory). We extend the rigity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion etale motives essentially coincide with the usual complexes of torsion etale sheaves (at least if we restrict ourselves to torsion prime to the residue characteristics). As a consequence, we obtain the expected results of absolute purity, of finiteness, and of Grothendieck duality for etale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion etale sheaves. Following Thomason's insights, this also provides a conceptual and convenient construction of the $\ell$-adic realization of motives, as the homotopy $\ell$-completion functor.