The Stacks project

107.12 Nonabelian category of quasi-coherent modules

In Sheaves on Stacks, Section 93.11 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 93.14 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 84.5.9. 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 93.14 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 93.11.6 that $\mathcal{F}$ is given by a sequence $(\mathcal{F}_ n)$ where

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

  2. 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 10.97.1. 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 107.9.

Lemma 107.12.1. The category of quasi-coherent1 modules on a formal algebraic space $X$ is not abelian in general, even if $X$ is a Noetherian affine formal algebraic space.

Proof. See discussion above. $\square$

[1] With quasi-coherent modules as defined above. Due to how things are setup in the Stacks project, this is really the correct definition; as seen above our definition agrees with what one would naively have defined to be quasi-coherent modules on $\text{Spf}(A)$, namely complete $A$-modules.

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