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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

1. 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 $\mathcal{O}_{\mathbb{P}^n}(\mathbb{P}^n) \cong \mathbb{Z}$.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeApr 30th 2015

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.

2. Nice! I’ll add both mnemonics to the article.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 26th 2018
• (edited Jul 26th 2018)

What’s a canonical citation for the equivalence between the category of affine schemes and the opposite of finitely generated reduced algebras?

(The reference [21] given on Wikipedia here seems to be broken. Following it, I find no prop. 2.3 at all in chapter II. (?))

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 26th 2018

Made explicit the fully faithfulness of $\mathcal{O} \colon Schemes_{Aff} \to Ring^{op}$ (here) and added pointer to Hartschorne’s “Algebraic Geometry”, chapter II, prop. 2.3 for proof.