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  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(SpecR,Y)CRing(𝒪 Y(Y),R)Sch(Spec R, Y) \cong CRing(\mathcal{O}_Y(Y), R)” is wrong. Take Y= nY = \mathbb{P}^n and R=R = \mathbb{Z}. Then the left hand side consists of all the \mathbb{Z}-valued points of n\mathbb{P}^n. On the other hand, the right hand side only contains the unique ring homomorphism \mathbb{Z} \to \mathbb{Z}, since 𝒪 n( n)\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 RP(X×Y)R \in P(X \times Y) is a relation, then the Galois connection it induces between PXP X and PYP Y looks like TS\RT \leq S \backslash R iff SR/TS \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. (?))

    diff, v10, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2018

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

    diff, v11, current

  3. Affine schemes are really isomorphic to prime spectra, not locally isomorphic; the statement would have been correct for general schemes qua locally ringed spaces.


    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 18th 2021

    Added redirect: Zariski duality. To satisfy a link at duality between geometry and algebra.

    diff, v13, current

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