## 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

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.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)