The Stacks project

Lemma 109.13.1. 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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