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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.
Added redirect: Zariski duality. To satisfy a link at duality between geometry and algebra.
I feel like some things could be stated in a stronger and more categorical way. I found some of what was written confusingly weak, but maybe I’m getting something wrong.
The stated bijection is natural [1], i.e. this is really an adjunction $\mathrm{CRing}(T, \Gamma_Y\mathcal O\Gamma \dashv \mathrm{Spec}$ between $\mathrm{CRing}^\mathrm{op}$ and Scheme. Furthermore not only exists there some fully faithful functor from the affine schemes to the category of rings as stated in Proposition 2.1, but Spec is fully faithful, i.e. $\mathrm{CRing}^\mahrm{op}$ is a full reflective subcategory of Schemes, equivalent to it’s image, the affine schemes 2.
Lastly the statement that $h : \mathrm{Scheme}(\mathrm{Spec}(-), -) : \mathrm{Scheme} \to \mathrm{PSh}(\mathrm{CRing})$ being fully faithful is the same as saying that Spec is a dense functor or that the affine schemes are a dense subcategory of the category of schemes. This might be nice to state explicitly, as I was struggling to find anything about whether this was the case, though, maybe it’s just not interesting.
[1]: Hartshorn, exercise II.2.4
Adding reference
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