Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
  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.

    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 ΓSpec\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(),):SchemePSh(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..
  4. Adding reference

    Anonymous

    diff, v14, current

  5. adding

    Anonymous

    diff, v14, current