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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=ℙ nY = \mathbb{P}^n and R=ℤR = \mathbb{Z}. Then the left hand side consists of all the ℤ\mathbb{Z}-valued points of ℙ n\mathbb{P}^n. On the other hand, the right hand side only contains the unique ring homomorphism ℤ→ℤ\mathbb{Z} \to \mathbb{Z}, since 𝒪 ℙ n(ℙ n)≅ℤ\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∈P(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 T≤S\RT \leq S \backslash R iff S≤R/TS \leq R/T. You’re always homming into (not out of) the functorial construction.

  3. 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 Aff→Ring 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

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

    Anonymous

    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

    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTimeApr 16th 2022

    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 Γ⊣Spec\mathrm{CRing}(T, \Gamma_Y\mathcal O\Gamma \dashv \mathrm{Spec} between CRing op\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. CRing mahrmop\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:Scheme(Spec(−),−):Scheme→PSh(CRing)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

    • CommentRowNumber9.
    • CommentAuthorjulia9367
    • CommentTimeApr 16th 2022
    Forgot to sign in for that post..

  10. Adding reference

    Anonymous

    diff, v14, current

  11. adding

    Anonymous

    diff, v14, current

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeMay 30th 2024
    • (edited May 30th 2024)

    In this note we give an exposition of the well-known results Gabriel, which show how to define affine schemes in terms of the theory of noncommutative localisation.

    diff, v15, current

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