Motivic Weight Complexes for Arithmetic Varieties

TL;DR

This paper extends the construction of weight complexes of motives to arithmetic varieties and stacks, providing new tools for understanding their Euler characteristics without relying on resolution of singularities.

Contribution

It introduces weight complexes for arithmetic varieties and stacks using K_0-motives, and proves contravariance of these complexes under certain morphisms, broadening their applicability.

Findings

Weight complexes are associated to arithmetic varieties and stacks.

Contravariance of weight complexes under finite tor-dimension morphisms is established.

Euler characteristics in the Grothendieck group of motives are derived for these varieties.

Abstract

We associate weight complexes of (homological) motives, and hence Euler characteristics in the Grothendieck group of motives, to arithmetic varieties and Deligne-Mumford stacks; this extends the results in the paper "Descent, Motives and K-theory" in volume 478 of Crelle, where a similar result was proved for varieties over a field of characteristic zero. We use K_0-motives with rational coefficients, rather than Chow motives, because we cannot appeal to resolution of singularities, but rather must use de Jong's results. In addition, for varieties over a field we prove a general result on contravariance of weight complexes, in particular showing that any morphism of finite tor-dimension between varieties induces a morphism of weight complexes.