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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
At affine scheme, the fundamental theorem on morphisms of schemes was stated the other way round. I fixed that.
As a handy mnemonic, here is a quick and down-to-earth way to see that the claim “$Sch(Spec R, Y) \cong CRing(\mathcal{O}_Y(Y), R)$” is wrong. Take $Y = \mathbb{P}^n$ and $R = \mathbb{Z}$. Then the left hand side consists of all the $\mathbb{Z}$-valued points of $\mathbb{P}^n$. On the other hand, the right hand side only contains the unique ring homomorphism $\mathbb{Z} \to \mathbb{Z}$, since $\mathcal{O}_{\mathbb{P}^n}(\mathbb{P}^n) \cong \mathbb{Z}$.
Another mnemonic is that such dualities work the same way as ordinary Galois connections between power sets. If $R \in P(X \times Y)$ is a relation, then the Galois connection it induces between $P X$ and $P Y$ looks like $T \leq S \backslash R$ iff $S \leq R/T$. You’re always homming into (not out of) the functorial construction.
Nice! I’ll add both mnemonics to the article.
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