| import Mathlib.AlgebraicGeometry.AffineSpace |
| import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| An algebra equivalence with a polynomial ring yields an isomorphism to affine |
| space **over the coefficient field**. The quotient presentation also supplies a |
| closed immersion, so neither assertion is encoded by a bare bijection of points. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace StableCoordinate |
| namespace AffineGeometry |
|
|
| open CategoryTheory AlgebraicGeometry |
| open scoped AlgebraicGeometry |
|
|
| universe u |
|
|
| variable {k A B : Type u} [CommRing k] [CommRing A] [CommRing B] |
| variable [Algebra k A] [Algebra k B] |
|
|
| /-- The spectrum of a `k`-algebra, viewed as a scheme over `Spec k`. -/ |
| def algebraSpec (k A : Type u) [CommRing k] [CommRing A] [Algebra k A] : |
| Over (Spec (CommRingCat.of k)) := |
| Over.mk (Spec.map (CommRingCat.ofHom (algebraMap k A))) |
|
|
| /-- Contravariance takes an algebra equivalence to an actual scheme isomorphism. -/ |
| def specIsoOfAlgEquiv (e : A ≃ₐ[k] B) : |
| Spec (CommRingCat.of B) ≅ Spec (CommRingCat.of A) := |
| Scheme.Spec.mapIso e.toRingEquiv.toCommRingCatIso.op |
|
|
| /-- This scheme isomorphism respects the coefficient base scheme. -/ |
| theorem specIsoOfAlgEquiv_hom_over (e : A ≃ₐ[k] B) : |
| (specIsoOfAlgEquiv e).hom ≫ Spec.map (CommRingCat.ofHom (algebraMap k A)) = |
| Spec.map (CommRingCat.ofHom (algebraMap k B)) := by |
| change Spec.map (CommRingCat.ofHom e.toRingHom) ≫ _ = _ |
| rw [← Spec.map_comp] |
| congr 1 |
| ext r |
| exact e.commutes r |
|
|
| /-- The stronger isomorphism in the category of schemes over `Spec k`. -/ |
| def algebraSpecIso (e : A ≃ₐ[k] B) : algebraSpec k B ≅ algebraSpec k A := |
| Over.isoMk (specIsoOfAlgEquiv e) (specIsoOfAlgEquiv_hom_over e) |
|
|
| /-- Full faithfulness of `Spec` recovers the ring map from every affine scheme isomorphism. -/ |
| def ringEquivOfSpecIso |
| (e : Spec (CommRingCat.of B) ≅ Spec (CommRingCat.of A)) : A ≃+* B := |
| (Scheme.Spec.preimageIso e).unop.commRingCatIsoToRingEquiv |
|
|
| /-- The recovered ring equivalence is exactly the pullback of the original morphism. -/ |
| theorem ringEquivOfSpecIso_apply |
| (e : Spec (CommRingCat.of B) ≅ Spec (CommRingCat.of A)) (a : A) : |
| ringEquivOfSpecIso e a = Spec.preimage e.hom a := rfl |
|
|
| /-- Every affine scheme isomorphism over the coefficient base induces an algebra |
| isomorphism. Thus algebra automorphisms do not omit any ambient automorphisms. -/ |
| def algEquivOfSpecIso |
| (e : Spec (CommRingCat.of B) ≅ Spec (CommRingCat.of A)) |
| (he : e.hom ≫ Spec.map (CommRingCat.ofHom (algebraMap k A)) = |
| Spec.map (CommRingCat.ofHom (algebraMap k B))) : A ≃ₐ[k] B := |
| AlgEquiv.ofRingEquiv (f := ringEquivOfSpecIso e) (by |
| intro r |
| have h := congrArg Spec.preimage he |
| rw [Spec.preimage_comp, Spec.preimage_map, Spec.preimage_map] at h |
| exact congrArg (fun f : CommRingCat.of k ⟶ CommRingCat.of B => f r) h) |
|
|
| /-- In particular, every automorphism in the category of schemes over `Spec k` |
| is accounted for by an algebra automorphism. -/ |
| def algEquivOfAlgebraSpecIso (e : algebraSpec k B ≅ algebraSpec k A) : A ≃ₐ[k] B := |
| algEquivOfSpecIso ((Over.forget _).mapIso e) e.hom.w |
|
|
| /-- Recovering the algebra map and applying `Spec` returns the given isomorphism. -/ |
| theorem specIsoOfAlgEquiv_algEquivOfSpecIso |
| (e : Spec (CommRingCat.of B) ≅ Spec (CommRingCat.of A)) |
| (he : e.hom ≫ Spec.map (CommRingCat.ofHom (algebraMap k A)) = |
| Spec.map (CommRingCat.ofHom (algebraMap k B))) : |
| specIsoOfAlgEquiv (algEquivOfSpecIso e he) = e := by |
| apply Iso.ext |
| change Spec.map (Spec.preimage e.hom) = e.hom |
| exact Spec.map_preimage e.hom |
|
|
| /-- The ring and scheme-over-base existence formulations are equivalent. -/ |
| theorem nonempty_algEquiv_iff_algebraSpecIso : |
| Nonempty (A ≃ₐ[k] B) ↔ Nonempty (algebraSpec k B ≅ algebraSpec k A) := |
| ⟨fun ⟨e⟩ => ⟨algebraSpecIso e⟩, fun ⟨e⟩ => ⟨algEquivOfAlgebraSpecIso e⟩⟩ |
|
|
| /-- Transporting a defining ideal transports its closed subset by the associated |
| ambient scheme isomorphism (with the contravariant direction made explicit). -/ |
| theorem specIso_image_zeroLocus_of_ideal_map |
| (e : A ≃ₐ[k] B) (I : Ideal A) (J : Ideal B) |
| (h : I.map e.toRingHom = J) : |
| (specIsoOfAlgEquiv e.symm).hom '' |
| PrimeSpectrum.zeroLocus (I : Set A) = PrimeSpectrum.zeroLocus (J : Set B) := by |
| change PrimeSpectrum.comap e.symm.toRingHom '' PrimeSpectrum.zeroLocus (I : Set A) = _ |
| rw [image_comap_zeroLocus_eq_zeroLocus_comap _ _ e.symm.surjective] |
| change PrimeSpectrum.zeroLocus (I.comap e.toRingEquiv.symm : Set B) = _ |
| rw [Ideal.comap_symm e.toRingEquiv] |
| exact congrArg (fun K : Ideal B => PrimeSpectrum.zeroLocus (K : Set B)) h |
|
|
| /-- Conversely, equality of the closed subsets under any ambient scheme |
| isomorphism over the base recovers equality of their radical defining ideals. -/ |
| theorem ideal_map_of_specIso_image_zeroLocus |
| (I : Ideal B) (J : Ideal A) (hI : I.IsRadical) (hJ : J.IsRadical) |
| (e : Spec (CommRingCat.of B) ≅ Spec (CommRingCat.of A)) |
| (he : e.hom ≫ Spec.map (CommRingCat.ofHom (algebraMap k A)) = |
| Spec.map (CommRingCat.ofHom (algebraMap k B))) |
| (h : e.hom '' PrimeSpectrum.zeroLocus (I : Set B) = |
| PrimeSpectrum.zeroLocus (J : Set A)) : |
| I.map (algEquivOfSpecIso e he).symm.toRingHom = J := by |
| let a := algEquivOfSpecIso e he |
| have ha : Spec.map (CommRingCat.ofHom a.toRingHom) = e.hom := by |
| exact congrArg Iso.hom (specIsoOfAlgEquiv_algEquivOfSpecIso e he) |
| have hz : PrimeSpectrum.zeroLocus (I.comap a.toRingHom : Set A) = |
| PrimeSpectrum.zeroLocus (J : Set A) := by |
| rw [← image_comap_zeroLocus_eq_zeroLocus_comap B a.toRingHom a.surjective I] |
| change Spec.map (CommRingCat.ofHom a.toRingHom) '' |
| PrimeSpectrum.zeroLocus (I : Set B) = _ |
| rw [ha] |
| exact h |
| have hi : I.comap a.toRingHom = J := by |
| have hz' := PrimeSpectrum.zeroLocus_eq_iff.mp hz |
| rwa [(hI.comap a.toRingHom).radical, hJ.radical] at hz' |
| change I.map a.toRingEquiv.symm = J |
| rw [Ideal.map_symm a.toRingEquiv] |
| exact hi |
|
|
| /-- For radical defining ideals, the algebraic orbit condition is equivalent to |
| the geometric orbit condition under **all** ambient automorphisms over the base. -/ |
| theorem ideal_transport_iff_scheme_transport (I J : Ideal A) |
| (hI : I.IsRadical) (hJ : J.IsRadical) : |
| (∃ e : A ≃ₐ[k] A, I.map e.toRingHom = J) ↔ |
| ∃ e : Spec (CommRingCat.of A) ≅ Spec (CommRingCat.of A), |
| (e.hom ≫ Spec.map (CommRingCat.ofHom (algebraMap k A)) = |
| Spec.map (CommRingCat.ofHom (algebraMap k A))) ∧ |
| e.hom '' PrimeSpectrum.zeroLocus (I : Set A) = |
| PrimeSpectrum.zeroLocus (J : Set A) := by |
| constructor |
| · rintro ⟨e, he⟩ |
| exact ⟨specIsoOfAlgEquiv e.symm, specIsoOfAlgEquiv_hom_over e.symm, |
| specIso_image_zeroLocus_of_ideal_map e I J he⟩ |
| · rintro ⟨e, he, h⟩ |
| exact ⟨(algEquivOfSpecIso e he).symm, |
| ideal_map_of_specIso_image_zeroLocus I J hI hJ e he h⟩ |
|
|
| /-- A polynomial-ring algebra presentation gives affine space as a scheme, |
| not just its underlying collection of rational points. -/ |
| def affineSpaceIsoOfAlgEquiv {σ : Type u} (e : A ≃ₐ[k] MvPolynomial σ k) : |
| Spec (CommRingCat.of A) ≅ AffineSpace σ (Spec (CommRingCat.of k)) := |
| specIsoOfAlgEquiv e.symm ≪≫ (AffineSpace.SpecIso σ (CommRingCat.of k)).symm |
|
|
| /-- The polynomial presentation isomorphism is an isomorphism over `Spec k`. -/ |
| theorem affineSpaceIsoOfAlgEquiv_hom_over {σ : Type u} |
| (e : A ≃ₐ[k] MvPolynomial σ k) : |
| (affineSpaceIsoOfAlgEquiv e).hom ≫ |
| (AffineSpace σ (Spec (CommRingCat.of k)) ↘ Spec (CommRingCat.of k)) = |
| Spec.map (CommRingCat.ofHom (algebraMap k A)) := by |
| simp only [affineSpaceIsoOfAlgEquiv, Iso.trans_hom, Iso.symm_hom, Category.assoc, |
| AffineSpace.SpecIso_inv_over] |
| exact specIsoOfAlgEquiv_hom_over e.symm |
|
|
| /-- Scheme-over-base version of polynomial-ring coordinates. -/ |
| def affineSpaceOverIsoOfAlgEquiv {σ : Type u} |
| (e : A ≃ₐ[k] MvPolynomial σ k) : |
| algebraSpec k A ≅ Over.mk |
| (AffineSpace σ (Spec (CommRingCat.of k)) ↘ Spec (CommRingCat.of k)) := |
| Over.isoMk (affineSpaceIsoOfAlgEquiv e) (affineSpaceIsoOfAlgEquiv_hom_over e) |
|
|
| /-- The natural map from the spectrum of a quotient into the ambient affine scheme. -/ |
| def quotientEmbedding (I : Ideal A) : |
| Spec (CommRingCat.of (A ⧸ I)) ⟶ Spec (CommRingCat.of A) := |
| Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I)) |
|
|
| /-- This is genuinely a closed immersion. -/ |
| instance quotientEmbedding_isClosedImmersion (I : Ideal A) : |
| IsClosedImmersion (quotientEmbedding I) := |
| IsClosedImmersion.spec_of_surjective _ Ideal.Quotient.mk_surjective |
|
|
| /-- The embedded image is exactly the closed subset defined by the ideal. -/ |
| theorem quotientEmbedding_range (I : Ideal A) : |
| Set.range (quotientEmbedding I) = PrimeSpectrum.zeroLocus (I : Set A) := by |
| change Set.range (PrimeSpectrum.comap (Ideal.Quotient.mk I)) = _ |
| rw [range_comap_of_surjective _ _ Ideal.Quotient.mk_surjective] |
| simp |
|
|
| /-- In particular every algebra equivalence of a hypersurface coordinate ring |
| with a polynomial algebra gives the required affine-space isomorphism. -/ |
| def hypersurfaceIsoOfAlgEquiv {σ τ : Type u} |
| (p : MvPolynomial σ k) |
| (e : (MvPolynomial σ k ⧸ Ideal.span {p}) ≃ₐ[k] MvPolynomial τ k) : |
| Spec (CommRingCat.of (MvPolynomial σ k ⧸ Ideal.span {p})) ≅ |
| AffineSpace τ (Spec (CommRingCat.of k)) := |
| affineSpaceIsoOfAlgEquiv e |
|
|
| end AffineGeometry |
| end StableCoordinate |
|
|
| end |
|
|
| end OAI |
|
|
