Let $X = $ Spec $A$ be an affine scheme. I know that there is an inclusion of categories from $A$-modules to sheaves of $\mathcal O_X$-modules on $X$, which is exact and fully faithful.

It seems to me that by Lemma 17.10.5./(4) of the stacks project (https://stacks.math.columbia.edu/tag/01BH), this inclusion functor has a right adjoint (the global section functor), hence $A$-modules form a reflective subcategory. The image of the inclusion functor is precisely the category of quasi-coherent sheaves on $X$.

My question is: does this generalize to arbitrary (non-affine) schemes? This would immediately give a nice proof that quasi-coherent sheaves on any scheme form an Abelian category.

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    $\begingroup$ Note that also in your example, you have a coreflective subcategory, not a reflective one. $\endgroup$ Sep 13, 2023 at 7:53

1 Answer 1


The category of quasicoherent sheaves is actually a coreflective subcategory of the category of all sheaves of modules on the scheme.

See the stacks project.

  • $\begingroup$ Dear Lukas. Indeed, my example gives a coreflective subcategory. I have no idea why I thought it was reflective. Thank you for the link! $\endgroup$
    – Adelhart
    Sep 13, 2023 at 14:23

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