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Cubical/AlgebraicGeometry/Functorial/ZFunctors/Base.agda
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| {- | ||
| A ℤ-functor is just a functor from rings to sets. | ||
| NOTE: we consider the functor category [ Ring ℓ , Set ℓ ] for some universe level ℓ | ||
| and not [ Ring ℓ , Set (ℓ+1) ] as is done in | ||
| "Introduction to Algebraic Geometry and Algebraic Groups" | ||
| by Demazure & Gabriel! | ||
| The category of ℤ-functors contains the category of (qcqs-) schemes | ||
| as a full subcategory and satisfies a "universal property" | ||
| similar to the one of schemes: | ||
| There is an adjunction 𝓞 ⊣ᵢ Sp | ||
| (relative to the inclusion i : CommRing ℓ → CommRing (ℓ+1)) | ||
| between the "global sections functor" 𝓞 | ||
| and the fully-faithful embedding of affines Sp, | ||
| whose counit is a natural isomorphism | ||
| -} | ||
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| {-# OPTIONS --safe --lossy-unification #-} | ||
| module Cubical.AlgebraicGeometry.Functorial.ZFunctors.Base where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.Equiv | ||
| open import Cubical.Foundations.Isomorphism | ||
| open import Cubical.Foundations.Function | ||
| open import Cubical.Foundations.Powerset | ||
| open import Cubical.Foundations.HLevels | ||
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| open import Cubical.Functions.FunExtEquiv | ||
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| open import Cubical.Data.Sigma | ||
| open import Cubical.Data.Nat using (ℕ) | ||
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| open import Cubical.Algebra.Ring | ||
| open import Cubical.Algebra.CommRing | ||
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| open import Cubical.Categories.Category renaming (isIso to isIsoC) | ||
| open import Cubical.Categories.Functor | ||
| open import Cubical.Categories.Instances.Sets | ||
| open import Cubical.Categories.Instances.CommRings | ||
| open import Cubical.Categories.Instances.Functors | ||
| open import Cubical.Categories.NaturalTransformation | ||
| open import Cubical.Categories.Yoneda | ||
| open import Cubical.Categories.Site.Sheaf | ||
| open import Cubical.Categories.Site.Instances.ZariskiCommRing | ||
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| open import Cubical.HITs.PropositionalTruncation as PT | ||
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| open Category hiding (_∘_) renaming (_⋆_ to _⋆c_) | ||
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| module _ {ℓ : Level} where | ||
|
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| open Functor | ||
| open NatTrans | ||
| open CommRingStr ⦃...⦄ | ||
| open IsRingHom | ||
|
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||
| -- using the naming conventions of Demazure & Gabriel | ||
| ℤFunctor = Functor (CommRingsCategory {ℓ = ℓ}) (SET ℓ) | ||
| ℤFUNCTOR = FUNCTOR (CommRingsCategory {ℓ = ℓ}) (SET ℓ) | ||
|
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| -- uses that double op is original category definitionally | ||
| Sp : Functor (CommRingsCategory {ℓ = ℓ} ^op) ℤFUNCTOR | ||
| Sp = YO {C = (CommRingsCategory {ℓ = ℓ} ^op)} | ||
|
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||
| -- Affine schemes are merely representable functors | ||
| -- TODO: by univalence it should be fine to take untruncated Σ | ||
| isAffine : ℤFunctor → Type (ℓ-suc ℓ) | ||
| isAffine X = ∃[ A ∈ CommRing ℓ ] NatIso (Sp .F-ob A) X | ||
|
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||
| -- a ℤ-functor that is a sheaf wrt the Zariski coverage is called local | ||
| isLocal : ℤFunctor → Type (ℓ-suc ℓ) | ||
| isLocal X = isSheaf zariskiCoverage X | ||
|
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||
| -- the forgetful functor | ||
| -- aka the affine line | ||
| -- (aka the representable of ℤ[x]) | ||
| 𝔸¹ : ℤFunctor | ||
| 𝔸¹ = ForgetfulCommRing→Set | ||
|
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| -- the "global sections" functor | ||
| 𝓞 : Functor ℤFUNCTOR (CommRingsCategory {ℓ = ℓ-suc ℓ} ^op) | ||
| fst (F-ob 𝓞 X) = X ⇒ 𝔸¹ | ||
|
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| -- ring struncture induced by internal ring object 𝔸¹ | ||
| N-ob (CommRingStr.0r (snd (F-ob 𝓞 X))) A _ = 0r | ||
| where instance _ = A .snd | ||
| N-hom (CommRingStr.0r (snd (F-ob 𝓞 X))) φ = funExt λ _ → sym (φ .snd .pres0) | ||
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| N-ob (CommRingStr.1r (snd (F-ob 𝓞 X))) A _ = 1r | ||
| where instance _ = A .snd | ||
| N-hom (CommRingStr.1r (snd (F-ob 𝓞 X))) φ = funExt λ _ → sym (φ .snd .pres1) | ||
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| N-ob ((snd (F-ob 𝓞 X) CommRingStr.+ α) β) A x = α .N-ob A x + β .N-ob A x | ||
| where instance _ = A .snd | ||
| N-hom ((snd (F-ob 𝓞 X) CommRingStr.+ α) β) {x = A} {y = B} φ = funExt path | ||
| where | ||
| instance | ||
| _ = A .snd | ||
| _ = B .snd | ||
| path : ∀ x → α .N-ob B (X .F-hom φ x) + β .N-ob B (X .F-hom φ x) | ||
| ≡ φ .fst (α .N-ob A x + β .N-ob A x) | ||
| path x = α .N-ob B (X .F-hom φ x) + β .N-ob B (X .F-hom φ x) | ||
| ≡⟨ cong₂ _+_ (funExt⁻ (α .N-hom φ) x) (funExt⁻ (β .N-hom φ) x) ⟩ | ||
| φ .fst (α .N-ob A x) + φ .fst (β .N-ob A x) | ||
| ≡⟨ sym (φ .snd .pres+ _ _) ⟩ | ||
| φ .fst (α .N-ob A x + β .N-ob A x) ∎ | ||
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| N-ob ((snd (F-ob 𝓞 X) CommRingStr.· α) β) A x = α .N-ob A x · β .N-ob A x | ||
| where instance _ = A .snd | ||
| N-hom ((snd (F-ob 𝓞 X) CommRingStr.· α) β) {x = A} {y = B} φ = funExt path | ||
| where | ||
| instance | ||
| _ = A .snd | ||
| _ = B .snd | ||
| path : ∀ x → α .N-ob B (X .F-hom φ x) · β .N-ob B (X .F-hom φ x) | ||
| ≡ φ .fst (α .N-ob A x · β .N-ob A x) | ||
| path x = α .N-ob B (X .F-hom φ x) · β .N-ob B (X .F-hom φ x) | ||
| ≡⟨ cong₂ _·_ (funExt⁻ (α .N-hom φ) x) (funExt⁻ (β .N-hom φ) x) ⟩ | ||
| φ .fst (α .N-ob A x) · φ .fst (β .N-ob A x) | ||
| ≡⟨ sym (φ .snd .pres· _ _) ⟩ | ||
| φ .fst (α .N-ob A x · β .N-ob A x) ∎ | ||
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| N-ob ((CommRingStr.- snd (F-ob 𝓞 X)) α) A x = - α .N-ob A x | ||
| where instance _ = A .snd | ||
| N-hom ((CommRingStr.- snd (F-ob 𝓞 X)) α) {x = A} {y = B} φ = funExt path | ||
| where | ||
| instance | ||
| _ = A .snd | ||
| _ = B .snd | ||
| path : ∀ x → - α .N-ob B (X .F-hom φ x) ≡ φ .fst (- α .N-ob A x) | ||
| path x = - α .N-ob B (X .F-hom φ x) ≡⟨ cong -_ (funExt⁻ (α .N-hom φ) x) ⟩ | ||
| - φ .fst (α .N-ob A x) ≡⟨ sym (φ .snd .pres- _) ⟩ | ||
| φ .fst (- α .N-ob A x) ∎ | ||
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| CommRingStr.isCommRing (snd (F-ob 𝓞 X)) = makeIsCommRing | ||
| isSetNatTrans | ||
| (λ _ _ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.+Assoc _ _ _)) | ||
| (λ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.+IdR _)) | ||
| (λ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.+InvR _)) | ||
| (λ _ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.+Comm _ _)) | ||
| (λ _ _ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.·Assoc _ _ _)) | ||
| (λ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.·IdR _)) | ||
| (λ _ _ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.·DistR+ _ _ _)) | ||
| (λ _ _ → makeNatTransPath (funExt₂ λ A _ → A .snd .CommRingStr.·Comm _ _)) | ||
|
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| -- action on natural transformations | ||
| fst (F-hom 𝓞 α) = α ●ᵛ_ | ||
| pres0 (snd (F-hom 𝓞 α)) = makeNatTransPath (funExt₂ λ _ _ → refl) | ||
| pres1 (snd (F-hom 𝓞 α)) = makeNatTransPath (funExt₂ λ _ _ → refl) | ||
| pres+ (snd (F-hom 𝓞 α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) | ||
| pres· (snd (F-hom 𝓞 α)) _ _ = makeNatTransPath (funExt₂ λ _ _ → refl) | ||
| pres- (snd (F-hom 𝓞 α)) _ = makeNatTransPath (funExt₂ λ _ _ → refl) | ||
|
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| -- functoriality of 𝓞 | ||
| F-id 𝓞 = RingHom≡ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) | ||
| F-seq 𝓞 _ _ = RingHom≡ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) | ||
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| -- we get a relative adjunction 𝓞 ⊣ᵢ Sp | ||
| -- with respect to the inclusion i : CommRing ℓ → CommRing (ℓ+1) | ||
| module AdjBij {ℓ : Level} where | ||
|
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| open Functor | ||
| open NatTrans | ||
| open Iso | ||
| open IsRingHom | ||
|
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| private module _ {A : CommRing ℓ} {X : ℤFunctor} where | ||
| _♭ : CommRingHom A (𝓞 .F-ob X) → X ⇒ Sp .F-ob A | ||
| fst (N-ob (φ ♭) B x) a = φ .fst a .N-ob B x | ||
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| pres0 (snd (N-ob (φ ♭) B x)) = cong (λ y → y .N-ob B x) (φ .snd .pres0) | ||
| pres1 (snd (N-ob (φ ♭) B x)) = cong (λ y → y .N-ob B x) (φ .snd .pres1) | ||
| pres+ (snd (N-ob (φ ♭) B x)) _ _ = cong (λ y → y .N-ob B x) (φ .snd .pres+ _ _) | ||
| pres· (snd (N-ob (φ ♭) B x)) _ _ = cong (λ y → y .N-ob B x) (φ .snd .pres· _ _) | ||
| pres- (snd (N-ob (φ ♭) B x)) _ = cong (λ y → y .N-ob B x) (φ .snd .pres- _) | ||
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| N-hom (φ ♭) ψ = funExt (λ x → RingHom≡ (funExt λ a → funExt⁻ (φ .fst a .N-hom ψ) x)) | ||
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| -- the other direction is just precomposition modulo Yoneda | ||
| _♯ : X ⇒ Sp .F-ob A → CommRingHom A (𝓞 .F-ob X) | ||
| fst (α ♯) a = α ●ᵛ yonedaᴾ 𝔸¹ A .inv a | ||
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| pres0 (snd (α ♯)) = makeNatTransPath (funExt₂ λ B x → α .N-ob B x .snd .pres0) | ||
| pres1 (snd (α ♯)) = makeNatTransPath (funExt₂ λ B x → α .N-ob B x .snd .pres1) | ||
| pres+ (snd (α ♯)) _ _ = makeNatTransPath (funExt₂ λ B x → α .N-ob B x .snd .pres+ _ _) | ||
| pres· (snd (α ♯)) _ _ = makeNatTransPath (funExt₂ λ B x → α .N-ob B x .snd .pres· _ _) | ||
| pres- (snd (α ♯)) _ = makeNatTransPath (funExt₂ λ B x → α .N-ob B x .snd .pres- _) | ||
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| -- the two maps are inverse to each other | ||
| ♭♯Id : ∀ (α : X ⇒ Sp .F-ob A) → ((α ♯) ♭) ≡ α | ||
| ♭♯Id _ = makeNatTransPath (funExt₂ λ _ _ → RingHom≡ (funExt (λ _ → refl))) | ||
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| ♯♭Id : ∀ (φ : CommRingHom A (𝓞 .F-ob X)) → ((φ ♭) ♯) ≡ φ | ||
| ♯♭Id _ = RingHom≡ (funExt λ _ → makeNatTransPath (funExt₂ λ _ _ → refl)) | ||
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| 𝓞⊣SpIso : {A : CommRing ℓ} {X : ℤFunctor {ℓ}} | ||
| → Iso (CommRingHom A (𝓞 .F-ob X)) (X ⇒ Sp .F-ob A) | ||
| fun 𝓞⊣SpIso = _♭ | ||
| inv 𝓞⊣SpIso = _♯ | ||
| rightInv 𝓞⊣SpIso = ♭♯Id | ||
| leftInv 𝓞⊣SpIso = ♯♭Id | ||
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| 𝓞⊣SpNatℤFunctor : {A : CommRing ℓ} {X Y : ℤFunctor {ℓ}} (α : X ⇒ Sp .F-ob A) (β : Y ⇒ X) | ||
| → (β ●ᵛ α) ♯ ≡ (𝓞 .F-hom β) ∘cr (α ♯) | ||
| 𝓞⊣SpNatℤFunctor _ _ = RingHom≡ (funExt (λ _ → makeNatTransPath (funExt₂ (λ _ _ → refl)))) | ||
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| 𝓞⊣SpNatCommRing : {X : ℤFunctor {ℓ}} {A B : CommRing ℓ} | ||
| (φ : CommRingHom A (𝓞 .F-ob X)) (ψ : CommRingHom B A) | ||
| → (φ ∘cr ψ) ♭ ≡ (φ ♭) ●ᵛ Sp .F-hom ψ | ||
| 𝓞⊣SpNatCommRing _ _ = makeNatTransPath (funExt₂ λ _ _ → RingHom≡ (funExt (λ _ → refl))) | ||
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| -- right relative adjunction induces a counit, | ||
| -- which is an equivalence | ||
| private | ||
| ε : (A : CommRing ℓ) → CommRingHom A ((𝓞 ∘F Sp) .F-ob A) | ||
| ε A = (idTrans (Sp .F-ob A)) ♯ | ||
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| 𝓞⊣SpCounitEquiv : (A : CommRing ℓ) → CommRingEquiv A ((𝓞 ∘F Sp) .F-ob A) | ||
| fst (𝓞⊣SpCounitEquiv A) = isoToEquiv theIso | ||
| where | ||
| theIso : Iso (A .fst) ((𝓞 ∘F Sp) .F-ob A .fst) | ||
| fun theIso = ε A .fst | ||
| inv theIso = yonedaᴾ 𝔸¹ A .fun | ||
| rightInv theIso α = ℤFUNCTOR .⋆IdL _ ∙ yonedaᴾ 𝔸¹ A .leftInv α | ||
| leftInv theIso a = path -- I get yellow otherwise | ||
| where | ||
| path : yonedaᴾ 𝔸¹ A .fun ((idTrans (Sp .F-ob A)) ●ᵛ yonedaᴾ 𝔸¹ A .inv a) ≡ a | ||
| path = cong (yonedaᴾ 𝔸¹ A .fun) (ℤFUNCTOR .⋆IdL _) ∙ yonedaᴾ 𝔸¹ A .rightInv a | ||
| snd (𝓞⊣SpCounitEquiv A) = ε A .snd |
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Cubical/AlgebraicGeometry/Functorial/ZFunctors/CompactOpen.agda
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Cubical/AlgebraicGeometry/Functorial/ZFunctors/OpenSubscheme.agda
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| {- | ||
| Compact open subfunctors of qcqs-schemes are qcqs-schemes (TODO!!!) | ||
| The proof proceeds by | ||
| 1. Defining standard/basic compact opens of affines and proving that they are affine | ||
| 2. Proving that arbitrary compact opens of affines a re qcqs-schemes | ||
| -} | ||
| {-# OPTIONS --safe --lossy-unification #-} | ||
| module Cubical.AlgebraicGeometry.Functorial.ZFunctors.OpenSubscheme where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.Equiv | ||
| open import Cubical.Foundations.Isomorphism | ||
| open import Cubical.Foundations.Function | ||
| open import Cubical.Foundations.Powerset | ||
| open import Cubical.Foundations.HLevels | ||
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| open import Cubical.Functions.FunExtEquiv | ||
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| open import Cubical.Data.Sigma | ||
| open import Cubical.Data.Nat using (ℕ) | ||
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| open import Cubical.Data.FinData | ||
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| open import Cubical.Algebra.Ring | ||
| open import Cubical.Algebra.CommRing | ||
| open import Cubical.Algebra.CommRing.Localisation | ||
| open import Cubical.Algebra.CommRing.RadicalIdeal | ||
| open import Cubical.Algebra.Semilattice | ||
| open import Cubical.Algebra.Lattice | ||
| open import Cubical.Algebra.DistLattice | ||
| open import Cubical.Algebra.DistLattice.BigOps | ||
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| open import Cubical.AlgebraicGeometry.ZariskiLattice.Base | ||
| open import Cubical.AlgebraicGeometry.ZariskiLattice.UniversalProperty | ||
| open import Cubical.AlgebraicGeometry.ZariskiLattice.Properties | ||
| open import Cubical.AlgebraicGeometry.Functorial.ZFunctors.Base | ||
| open import Cubical.AlgebraicGeometry.Functorial.ZFunctors.CompactOpen | ||
| open import Cubical.AlgebraicGeometry.Functorial.ZFunctors.QcQsScheme | ||
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| open import Cubical.Categories.Category renaming (isIso to isIsoC) | ||
| open import Cubical.Categories.Functor | ||
| open import Cubical.Categories.Instances.Sets | ||
| open import Cubical.Categories.Instances.CommRings | ||
| open import Cubical.Categories.Instances.DistLattice | ||
| open import Cubical.Categories.Instances.DistLattices | ||
| open import Cubical.Categories.Instances.Functors | ||
| open import Cubical.Categories.Site.Cover | ||
| open import Cubical.Categories.Site.Coverage | ||
| open import Cubical.Categories.Site.Sheaf | ||
| open import Cubical.Categories.Site.Instances.ZariskiCommRing | ||
| open import Cubical.Categories.NaturalTransformation | ||
| open import Cubical.Categories.Yoneda | ||
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| open import Cubical.HITs.PropositionalTruncation as PT | ||
| open import Cubical.HITs.SetQuotients as SQ | ||
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| open import Cubical.Relation.Binary.Order.Poset | ||
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| open Category hiding (_∘_) renaming (_⋆_ to _⋆c_) | ||
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| -- standard affine opens | ||
| module _ {ℓ : Level} (R : CommRing ℓ) (f : R .fst) where | ||
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| open Iso | ||
| open Functor | ||
| open NatTrans | ||
| open NatIso | ||
| open isIsoC | ||
| open DistLatticeStr ⦃...⦄ | ||
| open CommRingStr ⦃...⦄ | ||
| open IsRingHom | ||
| open RingHoms | ||
| open IsLatticeHom | ||
| open ZarLat | ||
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| open InvertingElementsBase R | ||
| open UniversalProp f | ||
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| private module ZL = ZarLatUniversalProp | ||
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| private instance _ = R .snd | ||
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| D : CompactOpen (Sp ⟅ R ⟆) | ||
| D = yonedaᴾ ZarLatFun R .inv (ZL.D R f) | ||
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| SpR[1/f]≅⟦Df⟧ : NatIso (Sp .F-ob R[1/ f ]AsCommRing) ⟦ D ⟧ᶜᵒ | ||
| N-ob (trans SpR[1/f]≅⟦Df⟧) B φ = (φ ∘r /1AsCommRingHom) , ∨lRid _ ∙ path | ||
| where | ||
| open CommRingHomTheory φ | ||
| open IsSupport (ZL.isSupportD B) | ||
| instance | ||
| _ = B .snd | ||
| _ = ZariskiLattice B .snd | ||
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| isUnitφ[f/1] : φ .fst (f /1) ∈ B ˣ | ||
| isUnitφ[f/1] = RingHomRespInv (f /1) ⦃ S/1⊆S⁻¹Rˣ f ∣ 1 , sym (·IdR f) ∣₁ ⦄ | ||
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| path : ZL.D B (φ .fst (f /1)) ≡ 1l | ||
| path = supportUnit _ isUnitφ[f/1] | ||
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| N-hom (trans SpR[1/f]≅⟦Df⟧) _ = funExt λ _ → Σ≡Prop (λ _ → squash/ _ _) (RingHom≡ refl) | ||
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| inv (nIso SpR[1/f]≅⟦Df⟧ B) (φ , Dφf≡D1) = invElemUniversalProp B φ isUnitφf .fst .fst | ||
| where | ||
| instance _ = ZariskiLattice B .snd | ||
| isUnitφf : φ .fst f ∈ B ˣ | ||
| isUnitφf = unitLemmaZarLat B (φ $r f) (sym (∨lRid _) ∙ Dφf≡D1) | ||
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| sec (nIso SpR[1/f]≅⟦Df⟧ B) = | ||
| funExt λ _ → Σ≡Prop (λ _ → squash/ _ _) (RingHom≡ (invElemUniversalProp _ _ _ .fst .snd)) | ||
| ret (nIso SpR[1/f]≅⟦Df⟧ B) = | ||
| funExt λ φ → cong fst (invElemUniversalProp B (φ ∘r /1AsCommRingHom) _ .snd (φ , refl)) | ||
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| isAffineD : isAffineCompactOpen D | ||
| isAffineD = ∣ R[1/ f ]AsCommRing , SpR[1/f]≅⟦Df⟧ ∣₁ | ||
|
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|
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| -- compact opens of affine schemes are qcqs-schemes | ||
| module _ {ℓ : Level} (R : CommRing ℓ) (W : CompactOpen (Sp ⟅ R ⟆)) where | ||
|
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| open Iso | ||
| open Functor | ||
| open NatTrans | ||
| open NatIso | ||
| open isIsoC | ||
| open DistLatticeStr ⦃...⦄ | ||
| open CommRingStr ⦃...⦄ | ||
| open PosetStr ⦃...⦄ | ||
| open IsRingHom | ||
| open RingHoms | ||
| open IsLatticeHom | ||
| open ZarLat | ||
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| open JoinSemilattice (Lattice→JoinSemilattice (DistLattice→Lattice (CompOpenDistLattice .F-ob (Sp .F-ob R)))) using (IndPoset; ind≤bigOp) | ||
| open InvertingElementsBase R | ||
| open Join | ||
| open JoinMap | ||
| open AffineCover | ||
| module ZL = ZarLatUniversalProp | ||
|
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| private | ||
| instance | ||
| _ = R .snd | ||
| _ = ZariskiLattice R .snd | ||
| _ = CompOpenDistLattice .F-ob (Sp .F-ob R) .snd | ||
| _ = CompOpenDistLattice .F-ob ⟦ W ⟧ᶜᵒ .snd | ||
| _ = IndPoset .snd | ||
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| w : ZL R | ||
| w = yonedaᴾ ZarLatFun R .fun W | ||
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| -- yoneda is a lattice homomorphsim | ||
| isHomYoneda : IsLatticeHom (DistLattice→Lattice (ZariskiLattice R) .snd) | ||
| (yonedaᴾ ZarLatFun R .inv) | ||
| (DistLattice→Lattice (CompOpenDistLattice ⟅ Sp ⟅ R ⟆ ⟆) .snd) | ||
| pres0 isHomYoneda = makeNatTransPath (funExt₂ (λ _ _ → refl)) | ||
| pres1 isHomYoneda = | ||
| makeNatTransPath (funExt₂ (λ _ φ → inducedZarLatHom φ .snd .pres1)) | ||
| pres∨l isHomYoneda u v = | ||
| makeNatTransPath (funExt₂ (λ _ φ → inducedZarLatHom φ .snd .pres∨l u v)) | ||
| pres∧l isHomYoneda u v = | ||
| makeNatTransPath (funExt₂ (λ _ φ → inducedZarLatHom φ .snd .pres∧l u v)) | ||
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| module _ {n : ℕ} | ||
| (f : FinVec (fst R) n) | ||
| (⋁Df≡W : ⋁ (CompOpenDistLattice ⟅ Sp ⟅ R ⟆ ⟆) (D R ∘ f) ≡ W) where | ||
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| Df≤W : ∀ i → D R (f i) ≤ W | ||
| Df≤W i = subst (D R (f i) ≤_) ⋁Df≡W (ind≤bigOp (D R ∘ f) i) | ||
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| toAffineCover : AffineCover ⟦ W ⟧ᶜᵒ | ||
| AffineCover.n toAffineCover = n | ||
| U toAffineCover i = compOpenDownHom (Sp ⟅ R ⟆) W .fst (D R (f i)) | ||
| covers toAffineCover = sym (pres⋁ (compOpenDownHom (Sp ⟅ R ⟆) W) (D R ∘ f)) | ||
| ∙ cong (compOpenDownHom (Sp ⟅ R ⟆) W .fst) ⋁Df≡W | ||
| ∙ makeNatTransPath (funExt₂ (λ _ → snd)) | ||
| isAffineU toAffineCover i = | ||
| ∣ _ , seqNatIso (SpR[1/f]≅⟦Df⟧ R (f i)) (compOpenDownHomNatIso _ (Df≤W i)) ∣₁ | ||
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| module _ {n : ℕ} | ||
| (f : FinVec (fst R) n) | ||
| (⋁Df≡w : ⋁ (ZariskiLattice R) (ZL.D R ∘ f) ≡ w) where | ||
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| private | ||
| ⋁Df≡W : ⋁ (CompOpenDistLattice ⟅ Sp ⟅ R ⟆ ⟆) (D R ∘ f) ≡ W | ||
| ⋁Df≡W = sym (pres⋁ (_ , isHomYoneda) (ZL.D R ∘ f)) | ||
| ∙ cong (yonedaᴾ ZarLatFun R .inv) ⋁Df≡w | ||
| ∙ yonedaᴾ ZarLatFun R .leftInv W | ||
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| makeAffineCoverCompOpenOfAffine : AffineCover ⟦ W ⟧ᶜᵒ | ||
| makeAffineCoverCompOpenOfAffine = toAffineCover f ⋁Df≡W | ||
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| hasAffineCoverCompOpenOfAffine : hasAffineCover ⟦ W ⟧ᶜᵒ | ||
| hasAffineCoverCompOpenOfAffine = PT.map truncHelper ([]surjective w) | ||
| where | ||
| truncHelper : Σ[ n,f ∈ Σ ℕ (FinVec (fst R)) ] [ n,f ] ≡ w → AffineCover ⟦ W ⟧ᶜᵒ | ||
| truncHelper ((n , f) , [n,f]≡w) = makeAffineCoverCompOpenOfAffine f (ZL.⋁D≡ R f ∙ [n,f]≡w) | ||
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| isQcQsSchemeCompOpenOfAffine : isQcQsScheme ⟦ W ⟧ᶜᵒ | ||
| fst isQcQsSchemeCompOpenOfAffine = presLocalCompactOpen _ _ (isSubcanonicalZariskiCoverage R) | ||
| snd isQcQsSchemeCompOpenOfAffine = hasAffineCoverCompOpenOfAffine |
97 changes: 97 additions & 0 deletions
97
Cubical/AlgebraicGeometry/Functorial/ZFunctors/QcQsScheme.agda
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| @@ -0,0 +1,97 @@ | ||
| {- | ||
| A qcqs-scheme is a ℤ-functor that is a local (a Zariski-sheaf) | ||
| and has an affine cover, where that notion of cover is given | ||
| by the lattice structure of compact open subfunctors | ||
| -} | ||
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| {-# OPTIONS --safe --lossy-unification #-} | ||
| module Cubical.AlgebraicGeometry.Functorial.ZFunctors.QcQsScheme where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.Equiv | ||
| open import Cubical.Foundations.Isomorphism | ||
| open import Cubical.Foundations.Function | ||
| open import Cubical.Foundations.Powerset | ||
| open import Cubical.Foundations.HLevels | ||
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| open import Cubical.Functions.FunExtEquiv | ||
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| open import Cubical.Data.Sigma | ||
| open import Cubical.Data.Nat using (ℕ) | ||
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| open import Cubical.Data.FinData | ||
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| open import Cubical.Algebra.Ring | ||
| open import Cubical.Algebra.CommRing | ||
| open import Cubical.Algebra.CommRing.Localisation | ||
| open import Cubical.Algebra.CommRing.RadicalIdeal | ||
| open import Cubical.Algebra.Semilattice | ||
| open import Cubical.Algebra.Lattice | ||
| open import Cubical.Algebra.DistLattice | ||
| open import Cubical.Algebra.DistLattice.BigOps | ||
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| open import Cubical.AlgebraicGeometry.Functorial.ZFunctors.Base | ||
| open import Cubical.AlgebraicGeometry.Functorial.ZFunctors.CompactOpen | ||
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| open import Cubical.Categories.Category renaming (isIso to isIsoC) | ||
| open import Cubical.Categories.Functor | ||
| open import Cubical.Categories.Instances.Sets | ||
| open import Cubical.Categories.Instances.CommRings | ||
| open import Cubical.Categories.Instances.DistLattice | ||
| open import Cubical.Categories.Instances.DistLattices | ||
| open import Cubical.Categories.Instances.Functors | ||
| open import Cubical.Categories.Site.Cover | ||
| open import Cubical.Categories.Site.Coverage | ||
| open import Cubical.Categories.Site.Sheaf | ||
| open import Cubical.Categories.Site.Instances.ZariskiCommRing | ||
| open import Cubical.Categories.NaturalTransformation | ||
| open import Cubical.Categories.Yoneda | ||
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| open import Cubical.HITs.PropositionalTruncation as PT | ||
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| open import Cubical.Relation.Binary.Order.Poset | ||
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| open Category hiding (_∘_) renaming (_⋆_ to _⋆c_) | ||
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| module _ {ℓ : Level} (X : ℤFunctor {ℓ = ℓ}) where | ||
| open Functor | ||
| open DistLatticeStr ⦃...⦄ | ||
| open Join (CompOpenDistLattice .F-ob X) | ||
| open JoinSemilattice (Lattice→JoinSemilattice (DistLattice→Lattice (CompOpenDistLattice .F-ob X))) | ||
| private instance _ = (CompOpenDistLattice .F-ob X) .snd | ||
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| record AffineCover : Type (ℓ-suc ℓ) where | ||
| field | ||
| n : ℕ | ||
| U : FinVec (CompactOpen X) n | ||
| covers : ⋁ U ≡ 1l -- TODO: equivalent to X ≡ ⟦ ⋁ U ⟧ᶜᵒ | ||
| isAffineU : ∀ i → isAffineCompactOpen (U i) | ||
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| hasAffineCover : Type (ℓ-suc ℓ) | ||
| hasAffineCover = ∥ AffineCover ∥₁ | ||
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| isQcQsScheme : Type (ℓ-suc ℓ) | ||
| isQcQsScheme = isLocal X × hasAffineCover | ||
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| -- affine schemes are qcqs-schemes | ||
| module _ {ℓ : Level} (A : CommRing ℓ) where | ||
| open Functor | ||
| open DistLatticeStr ⦃...⦄ | ||
| open AffineCover | ||
| private instance _ = (CompOpenDistLattice .F-ob (Sp .F-ob A)) .snd | ||
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| -- the canonical one element affine cover of a representable | ||
| singlAffineCover : AffineCover (Sp .F-ob A) | ||
| n singlAffineCover = 1 | ||
| U singlAffineCover zero = 1l | ||
| covers singlAffineCover = ∨lRid _ | ||
| isAffineU singlAffineCover zero = ∣ A , X≅⟦1⟧ (Sp .F-ob A) ∣₁ | ||
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| isQcQsSchemeAffine : isQcQsScheme (Sp .F-ob A) | ||
| fst isQcQsSchemeAffine = isSubcanonicalZariskiCoverage A | ||
| snd isQcQsSchemeAffine = ∣ singlAffineCover ∣₁ |
2 changes: 1 addition & 1 deletion
2
Cubical/Algebra/ZariskiLattice/Base.agda → ...lgebraicGeometry/ZariskiLattice/Base.agda
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