[Merged by Bors] - feat(AlgebraicGeometry/ProjectiveSpectrum/Scheme): finish the Proj construction
#12371
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Could you add an extended PR description (which would give an overall understanding of the last few PRs on related subjects), so that it appears in the |
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| /-- | ||
| This is the scheme `Proj(A)` for any `ℕ`-graded ring `A`. | ||
| -/ | ||
| def «Proj» : Scheme where |
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note that the french quotation can be safely ignored, they are here because Proj is a local notation. But since the notation is local, outside this file, one can just use AlgebraicGeometry.Proj
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Thanks very much! bors merge |
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…construction (#12371) This PR finishes the `Proj` construction for any ring graded by natural numbers: If $A$ is an $R$ algebra with 𝒜 as its grading i.e. 𝒜 i is an $R$-submodule of $A$ , then `AlgebraicGeometry.Proj 𝒜` is the scheme whose underlying topological space is the collection of homogeneous relevant prime ideals with the Zariski topology; whose sheaf of rings is the collection of functions that are locally homogeneous fractions. We prove that for each `f : A` with positive degree, `Proj | D(f)` (`Proj` as locally ringed space restricted to the basic open set around `f`) is isomorphic to `Spec A^0_f` (prime spectrum of the homogeneous localization of `A` at `f`). The isomorphism is constructed as the following: by the Gamma-Spec adjunction, it is sufficient to construct a ring map `A⁰_f → Γ(Proj, pbo f)` from the ring of homogeneous localization of `A` away from `f` to the local sections of structure sheaf of projective spectrum on the basic open set around `f`. The map `A⁰_f → Γ(Proj, pbo f)` is defined by sending `s ∈ A⁰_f` to the section `x ↦ s` on `pbo f`. Then we show that the map `Proj | D(f) -> Spec A⁰_f` induces an isomorphism on stalk level, thus is an isomorphism. Co-authored-by: Andrew Yang <[email protected]> Co-authored-by: zjj <[email protected]> Co-authored-by: Andrew Yang <[email protected]> Co-authored-by: Andrew Yang <[email protected]>
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Proj constructionProj construction
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…construction (#12371) This PR finishes the `Proj` construction for any ring graded by natural numbers: If $A$ is an $R$ algebra with 𝒜 as its grading i.e. 𝒜 i is an $R$-submodule of $A$ , then `AlgebraicGeometry.Proj 𝒜` is the scheme whose underlying topological space is the collection of homogeneous relevant prime ideals with the Zariski topology; whose sheaf of rings is the collection of functions that are locally homogeneous fractions. We prove that for each `f : A` with positive degree, `Proj | D(f)` (`Proj` as locally ringed space restricted to the basic open set around `f`) is isomorphic to `Spec A^0_f` (prime spectrum of the homogeneous localization of `A` at `f`). The isomorphism is constructed as the following: by the Gamma-Spec adjunction, it is sufficient to construct a ring map `A⁰_f → Γ(Proj, pbo f)` from the ring of homogeneous localization of `A` away from `f` to the local sections of structure sheaf of projective spectrum on the basic open set around `f`. The map `A⁰_f → Γ(Proj, pbo f)` is defined by sending `s ∈ A⁰_f` to the section `x ↦ s` on `pbo f`. Then we show that the map `Proj | D(f) -> Spec A⁰_f` induces an isomorphism on stalk level, thus is an isomorphism. Co-authored-by: Andrew Yang <[email protected]> Co-authored-by: zjj <[email protected]> Co-authored-by: Andrew Yang <[email protected]> Co-authored-by: Andrew Yang <[email protected]>
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…construction (#12371) This PR finishes the `Proj` construction for any ring graded by natural numbers: If $A$ is an $R$ algebra with 𝒜 as its grading i.e. 𝒜 i is an $R$-submodule of $A$ , then `AlgebraicGeometry.Proj 𝒜` is the scheme whose underlying topological space is the collection of homogeneous relevant prime ideals with the Zariski topology; whose sheaf of rings is the collection of functions that are locally homogeneous fractions. We prove that for each `f : A` with positive degree, `Proj | D(f)` (`Proj` as locally ringed space restricted to the basic open set around `f`) is isomorphic to `Spec A^0_f` (prime spectrum of the homogeneous localization of `A` at `f`). The isomorphism is constructed as the following: by the Gamma-Spec adjunction, it is sufficient to construct a ring map `A⁰_f → Γ(Proj, pbo f)` from the ring of homogeneous localization of `A` away from `f` to the local sections of structure sheaf of projective spectrum on the basic open set around `f`. The map `A⁰_f → Γ(Proj, pbo f)` is defined by sending `s ∈ A⁰_f` to the section `x ↦ s` on `pbo f`. Then we show that the map `Proj | D(f) -> Spec A⁰_f` induces an isomorphism on stalk level, thus is an isomorphism. Co-authored-by: Andrew Yang <[email protected]> Co-authored-by: zjj <[email protected]> Co-authored-by: Andrew Yang <[email protected]> Co-authored-by: Andrew Yang <[email protected]>
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This PR finishes the$A$ is an $R$ algebra with 𝒜 as its grading i.e. 𝒜 i is an $R$ -submodule of $A$ , then
Projconstruction for any ring graded by natural numbers:If
AlgebraicGeometry.Proj 𝒜is the scheme whose underlying topological space is the collection of homogeneous relevant prime ideals with the Zariski topology; whose sheaf of rings is the collection of functions that are locally homogeneous fractions. We prove that for eachf : Awith positive degree,Proj | D(f)(Projas locally ringed space restricted to the basic open set aroundf) is isomorphic toSpec A^0_f(prime spectrum of the homogeneous localization ofAatf).The isomorphism is constructed as the following:
by the Gamma-Spec adjunction, it is sufficient to construct a ring map
A⁰_f → Γ(Proj, pbo f)from the ring of homogeneous localization ofAaway fromfto the local sections of structure sheaf of projective spectrum on the basic open set aroundf.The map
A⁰_f → Γ(Proj, pbo f)is defined by sendings ∈ A⁰_fto the sectionx ↦ sonpbo f.Then we show that the map
Proj | D(f) -> Spec A⁰_finduces an isomorphism on stalk level, thus is an isomorphism.Co-authored-by: Andrew Yang [email protected]
HomogeneousLocalization.map#13332toSpecmap #13896IsLocalization#13933