The Stacks project

46.1 Introduction

For any scheme $X$ the category $\mathit{QCoh}(\mathcal{O}_ X)$ of quasi-coherent modules is abelian and a weak Serre subcategory of the abelian category of all $\mathcal{O}_ X$-modules. The same thing works for the category of quasi-coherent modules on an algebraic space $X$ viewed as a subcategory of the category of all $\mathcal{O}_ X$-modules on the small étale site of $X$. Moreover, for a quasi-compact and quasi-separated morphism $f : X \to Y$ the pushforward $f_*$ and higher direct images preserve quasi-coherence.

Next, let $X$ be a scheme and let $\mathcal{O}$ be the structure sheaf on one of the big sites of $X$, say, the big fppf site. The category of quasi-coherent $\mathcal{O}$-modules is abelian (in fact it is equivalent to the category of usual quasi-coherent $\mathcal{O}_ X$-modules on the scheme $X$ we mentioned above) but its imbedding into $\textit{Mod}(\mathcal{O})$ is not exact. An example is the map of quasi-coherent modules

\[ \mathcal{O}_{\mathbf{A}^1_ k} \longrightarrow \mathcal{O}_{\mathbf{A}^1_ k} \]

on $\mathbf{A}^1_ k = \mathop{\mathrm{Spec}}(k[x])$ given by multiplication by $x$. In the abelian category of quasi-coherent sheaves this map is injective, whereas in the abelian category of all $\mathcal{O}$-modules on the big site of $\mathbf{A}^1_ k$ this map has a nontrivial kernel as we see by evaluating on sections over $\mathop{\mathrm{Spec}}(k[x]/(x)) = \mathop{\mathrm{Spec}}(k)$. Moreover, for a quasi-compact and quasi-separated morphism $f : X \to Y$ the functor $f_{big, *}$ does not preserve quasi-coherence.

In this chapter we introduce the category of what we will call adequate modules, closely related to quasi-coherent modules, which “fixes” the two problems mentioned above. Another solution, which we will implement when we talk about quasi-coherent modules on algebraic stacks, is to consider $\mathcal{O}$-modules which are locally quasi-coherent and satisfy the flat base change property. See Cohomology of Stacks, Section 103.8, Cohomology of Stacks, Remark 103.10.7, and Derived Categories of Stacks, Section 104.5.

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