Lemma 109.13.1. The category of quasi-coherent^{1} modules on a formal algebraic space $X$ is not abelian in general, even if $X$ is a Noetherian affine formal algebraic space.

## 109.13 Nonabelian category of quasi-coherent modules

In Sheaves on Stacks, Section 95.11 we defined the category of quasi-coherent modules on a category fibred in groupoids over $\mathit{Sch}$. Although we show in Sheaves on Stacks, Section 95.15 that this category is abelian for algebraic stacks, in this section we show that this is not the case for formal algebraic spaces.

Namely, consider $\mathbf{Z}_ p$ viewed as topological ring using the $p$-adic topology. Let $X = \text{Spf}(\mathbf{Z}_ p)$, see Formal Spaces, Definition 86.9.9. Then $X$ is a sheaf in sets on $(\mathit{Sch}/\mathbf{Z})_{fppf}$ and gives rise to a stack in setoids $\mathcal{X}$, see Stacks, Lemma 8.6.2. Thus the discussion of Sheaves on Stacks, Section 95.15 applies.

Let $\mathcal{F}$ be a quasi-coherent module on $\mathcal{X}$. Since $X = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(\mathbf{Z}/p^ n\mathbf{Z})$ it is clear from Sheaves on Stacks, Lemma 95.12.2 that $\mathcal{F}$ is given by a sequence $(\mathcal{F}_ n)$ where

$\mathcal{F}_ n$ is a quasi-coherent module on $\mathop{\mathrm{Spec}}(\mathbf{Z}/p^ n\mathbf{Z})$, and

the transition maps give isomorphisms $\mathcal{F}_ n = \mathcal{F}_{n + 1}/p^ n\mathcal{F}_{n + 1}$.

Converting into modules we see that $\mathcal{F}$ corresponds to a system $(M_ n)$ where each $M_ n$ is an abelian group annihilated by $p^ n$ and the transition maps induce isomorphisms $M_ n = M_{n + 1}/p^ n M_{n + 1}$. In this situation the module $M = \mathop{\mathrm{lim}}\nolimits M_ n$ is a $p$-adically complete module and $M_ n = M/p^ n M$, see Algebra, Lemma 10.98.2. We conclude that the category of quasi-coherent modules on $X$ is equivalent to the category of $p$-adically complete abelian groups. This category is not abelian, see Section 109.10.

**Proof.**
See discussion above.
$\square$

## 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Your email address will not be published.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)