Sur la cat\'egorie d\'eriv\'ee des faisceaux tordus

TL;DR

This paper proves that the derived category of twisted quasi-coherent sheaves on a quasi-compact separated scheme is generated by objects supported on affine open subsets, extending known results to the twisted case.

Contribution

It establishes that the smallest triangulated subcategory containing all affine open immersion pushforwards generates the entire derived category of twisted quasi-coherent sheaves.

Findings

The subcategory $ ext{Delta}(X, ext{alpha})$ equals $D(QCoh(X, ext{alpha}))$.

The result extends to twisted sheaves a known generation property.

Provides a foundation for further studies on derived categories of twisted sheaves.

Abstract

Soit $X$ un sch\'ema quasi-compact et s\'epar\'e et soit $\alpha\in \check{C}^2(X, \mathcal O_X^*)$ un cocycle de C\v{e}ch. Nous consid\'erons la cat\'egorie d\'eriv\'ee $D(QCoh(X,\alpha))$ des faisceaux quasi-coh\'erents sur $X$ tordu par $\alpha$. Soit $\Delta(X,\alpha)$ la plus petite sous-cat\'egorie triangul\'ee de $D(QCoh(X,\alpha))$ contenant tous les objets $u_*\mathcal F^\bullet$, o\`u $u:U\longrightarrow X$ est une immersion ouverte avec $U$ affine et $\mathcal F^\bullet\in D(QCoh(U,u^*(\alpha)))$. Alors, le but de cet article est de montrer que $\Delta(X,\alpha)=D(QCoh(X,\alpha))$. -- Let $X$ be a quasi-compact and separated scheme and let $\alpha\in \check{C}^2(X, \mathcal O_X^*)$ be a C\v{e}ch cocycle. We consider the derived category $D(QCoh(X,\alpha))$ of quasi-coherent sheaves on $X$ twisted by $\alpha$. Let $\Delta(X,\alpha)$ be the smallest triangulated subcategory of $D(QCoh(X,\alpha))$ containing all the objects $u_*\mathcal F^\bullet$, where $u:U\longrightarrow X$ is an open immersion with $U$ affine and $\mathcal F^\bullet\in D(QCoh(U,u^*(\alpha)))$. Then, the purpose of this article is to show that $\Delta(X,\alpha)=D(QCoh(X,\alpha))$.