Faisceaux Q/Z(j) et conjecture de Gersten sur un corps imparfait

TL;DR

This paper reviews the properties of étale sheaf complexes $ ext{Q}/ ext{Z}(j)$ on schemes, providing a reference for the Gersten conjecture over imperfect fields and comparing specific cohomology groups.

Contribution

It offers a detailed reference for the properties of $ ext{Q}/ ext{Z}(j)$ sheaves and clarifies the Gersten conjecture and cohomology group comparisons over imperfect fields.

Findings

Explicit construction and properties of $ ext{Q}/ ext{Z}(j)$ sheaves are detailed.

Provides a precise reference for the Gersten conjecture over imperfect fields.

Clarifies the relationship between $ ext{H}^2( ext{--}, ext{Q}/ ext{Z}(1))$ and $ ext{H}^2( ext{--}, ext{G}_m)$.

Abstract

On revoit explicitement la construction ainsi que certaines propri\'et\'es des complexes de faisceaux \'etales $\mathbb{Q}/\mathbb{Z}(j)$ sur certains sch\'emas. Le but de ces notes est d'avoir une r\'ef\'erence pr\'ecise pour la conjecture de Gersten pour les faisceaux $\mathbb{Q}/\mathbb{Z}(j)$ sur un corps imparfait ainsi que pour la comparaison entre les groupes de cohomologie $\mathrm{H}^2(\ast,\mathbb{Q}/\mathbb{Z}(1))$ et $\mathrm{H}^2(\ast,\mathbb{G}_m)$. We review explicitly the definition and some properties of \'etale sheaf complexes $\mathbb{Q}/\mathbb{Z}(j)$ on some schemes. The purpose of these notes is to be a precise reference for the Gersten conjecture of $\mathbb{Q}/\mathbb{Z}(j)$ over an imperfect field and also for the comparison between the cohomology groups $\mathrm{H}^2(\ast,\mathbb{Q}/\mathbb{Z}(1))$ and $\mathrm{H}^2(\ast,\mathbb{G}_m)$.