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Question: Let $\mathcal A$ be an abelian category. Suppose that the only Serre subcategories of $\mathcal A$ are the zero category and $\mathcal A$ itself. Does it follow that every short exact sequence splits in $\mathcal A$?

Notes:

  • The hypothesis of no Serre subcategories means that the locally coherent Grothendieck category $Ind(\mathcal A)$ has trivial Ziegler spectrum.

  • The conclusion that short exact sequences in $\mathcal A$ split is not quite as strong as saying that $A$ is semisimple.

  • If $0 \to A \to B \to C \to 0$ is a non-split short exact sequence, then there is at least a weak Serre class given by $\{X \mid Ext^\ast(C,X) = 0\}$ which is not all of $\mathcal A$ (though maybe it could still be zero). But I'm not sure how to get an actual Serre class out of this.

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The category of finite abelian $p$-groups (where $p$ is your favourite prime) is an abelian category with no proper nonzero Serre subcategories, but not every short exact sequence splits.

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    $\begingroup$ Or more generally, take the category of finite length modules for a non-semisimple local ring. $\endgroup$
    – Jeremy Rickard
    Commented Dec 16, 2023 at 21:20

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