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This ought to be a comment, but I don't have the necessary reputation. I believe the claim is indeed false unless $M$ is locally of finite type. Here's a bit of context: It's always the case that an …

I assume that you mean that $\mathcal{F}$ is a sheaf of rings. What's internally an ideal of $\mathcal{F}$ is externally simply a sheaf of ideals. In case that the topos in question is the little Za …

We have a canonical map in one direction, namely $f^*(W(p)) \to W(f^*(p))$, but this map can fail to be an isomorphism. Here is an explicit counterexample. Let $X$ be the set of countably-brancing tre …

Here is a slightly different argument than algori's, not using the construction of the Čech-to-derived functor spectral sequence and only using $E_2$ terms, not $E_1$ terms. As you say, the spectral …

Unfortunately I don't know an interesting intrinsically formulated sufficient criterion for a locally ringed topos to be the little Zariski topos of a scheme. This is an extremely interesting question …

This question was previously asked over at math.SE. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of sets on $X$. Then we can define $\mathcal{O}_X\langle\mathcal{E}\rangle$, the free module over …

Great question! One example is Zorn's lemma. Assuming ZL holds in the metatheory, ZL also holds in toposes of sheaves over locales, so in particular in toposes of sheaves over topological spaces. Howe …