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fix whitespace, modify definition to use updated definition of ≼
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Cubical/Algebra/CommRing/RadicalIdeal.agda

Lines changed: 1 addition & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -240,7 +240,7 @@ module RadicalIdeal (R' : CommRing ℓ) where
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√·Absorb+ I J = CommIdeal≡Char (√·Absorb+LIncl I J) (√·Absorb+RIncl I J)
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√·ContrLIncl : (I J : CommIdeal) √ (√ I ·i √ J) ⊆ √ (I ·i J)
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√·ContrLIncl I J x = √·RContrLIncl I J x ∘ subst (x ∈_) (√·LContr I (√ J))
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√·ContrLIncl I J x = √·RContrLIncl I J x ∘ subst (x ∈_) (√·LContr I (√ J))
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√Pres·LIncl : (I J : CommIdeal) (√ I ·i √ J) ⊆ √ (I ·i J)
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√Pres·LIncl I J x = √·ContrLIncl I J x ∘ ∈→∈√ (√ I ·i √ J) x

Cubical/AlgebraicGeometry/ZariskiLattice/Base.agda

Lines changed: 5 additions & 5 deletions
Original file line numberDiff line numberDiff line change
@@ -155,15 +155,15 @@ module ZarLat (R' : CommRing ℓ) where
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∨a-join (_ , α) (_ , β) (_ , γ) = propBiimpl→Equiv (≼PropValued _ _) (isProp× (≼PropValued _ _) (≼PropValued _ _)) to fo
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where
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to : _
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to α∨β⊆γ .fst = ⊆-trans _ _ _
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to α∨β⊆γ .fst = ⊆-trans _ _ _
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(√Resp⊆ ⟨ α ⟩ ⟨ α ++Fin β ⟩ (⊆-trans _ _ _ (+iLincl ⟨ α ⟩ ⟨ β ⟩) (FGIdealAddLemmaRIncl _ α β))) α∨β⊆γ
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to α∨β⊆γ .snd = ⊆-trans _ _ _
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to α∨β⊆γ .snd = ⊆-trans _ _ _
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(√Resp⊆ ⟨ β ⟩ ⟨ α ++Fin β ⟩ (⊆-trans _ _ _ (+iRincl ⟨ α ⟩ ⟨ β ⟩) (FGIdealAddLemmaRIncl _ α β))) α∨β⊆γ
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fo : _
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fo (α⊆γ , β⊆γ) = ⊆-trans _ _ _ (√Resp⊆ ⟨ α ++Fin β ⟩ (⟨ α ⟩ +i ⟨ β ⟩) $ FGIdealAddLemmaLIncl _ α β) λ x x∈√α+β do
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(n , x^n∈α+β) x∈√α+β
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((y , z) , y∈α , z∈β , x^n≡y+z) x^n∈α+β
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^∈√→∈√ _ x n $ subst-∈ (√ ⟨ γ ⟩) (sym x^n≡y+z) $
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^∈√→∈√ _ x n $ subst-∈ (√ ⟨ γ ⟩) (sym x^n≡y+z) $
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√ ⟨ γ ⟩ .snd .+Closed (α⊆γ y (∈→∈√ _ y y∈α)) (β⊆γ z (∈→∈√ _ z z∈β))
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ZL-joins : isJoinSemipseudolattice ZLPoset
@@ -184,9 +184,9 @@ module ZarLat (R' : CommRing ℓ) where
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to γ≼α∧β .snd = ⊆-trans _ _ _ γ≼α∧β $ √Resp⊆ ⟨ α ··Fin β ⟩ ⟨ β ⟩ $
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⊆-trans _ _ _ (FGIdealMultLemmaLIncl _ α β) (·iRincl ⟨ α ⟩ ⟨ β ⟩)
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fo : _
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fo (γ≼α , γ≼β) = ⊆-trans _ _ _ (λ x x∈√γ
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fo (γ≼α , γ≼β) = ⊆-trans _ _ _ (λ x x∈√γ
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2 , subst-∈ (√ ⟨ α ⟩ ·i √ ⟨ β ⟩) (solve! R') (prodInProd _ _ x x (γ≼α x x∈√γ) (γ≼β x x∈√γ)) ∣₁
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) $ ⊆-trans _ _ _ (√·ContrLIncl ⟨ α ⟩ ⟨ β ⟩) $
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) $ ⊆-trans _ _ _ (√·ContrLIncl ⟨ α ⟩ ⟨ β ⟩) $
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√Resp⊆ (⟨ α ⟩ ·i ⟨ β ⟩) ⟨ α ··Fin β ⟩ $ FGIdealMultLemmaRIncl _ α β
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ZL-meets : isMeetSemipseudolattice ZLPoset

Cubical/AlgebraicGeometry/ZariskiLattice/Properties.agda

Lines changed: 3 additions & 4 deletions
Original file line numberDiff line numberDiff line change
@@ -65,7 +65,7 @@ module _ (R : CommRing ℓ) where
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D1≤Df = subst (_≤ D f) Df≡D1 (is-refl _)
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1∈√⟨f⟩ : 1r ∈ √ ⟨ replicateFinVec 1 f ⟩
68-
1∈√⟨f⟩ = isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun D1≤Df .fst zero
68+
1∈√⟨f⟩ = √FGIdealCharLImpl _ _ (isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun D1≤Df .fst) zero
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module _ {n : ℕ} (f₁,⋯,fₙ : FinVec (fst R) n) where
@@ -80,9 +80,8 @@ module _ (R : CommRing ℓ) where
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D1≤D[f₁,⋯,fₙ] = subst (_≤ D[f₁,⋯,fₙ]) D[f₁,⋯,fₙ]≡D1 (is-refl _)
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1∈√⟨f₁,⋯,fₙ⟩ : 1r ∈ √ ⟨ f₁,⋯,fₙ ⟩
83-
1∈√⟨f₁,⋯,fₙ⟩ = isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun
84-
D1≤D[f₁,⋯,fₙ] .fst zero
85-
83+
1∈√⟨f₁,⋯,fₙ⟩ = √FGIdealCharLImpl _ _
84+
(isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun D1≤D[f₁,⋯,fₙ] .fst) zero
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module LocDownSetIso (R : CommRing ℓ) (f : R .fst) where

Cubical/AlgebraicGeometry/ZariskiLattice/StructureSheaf.agda

Lines changed: 4 additions & 4 deletions
Original file line numberDiff line numberDiff line change
@@ -142,7 +142,7 @@ module _ {ℓ : Level} (R' : CommRing ℓ) where
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_ = snd R[1/ f ]AsCommRing
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f∈√⟨g⟩ : f ∈ √ ⟨ g ⟩ₛ
145-
f∈√⟨g⟩ = isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun Df≤Dg .fst zero
145+
f∈√⟨g⟩ = √FGIdealCharLImpl _ _ (isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun Df≤Dg .fst) zero
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-- The structure presheaf on BO
@@ -251,11 +251,11 @@ module _ {ℓ : Level} (R' : CommRing ℓ) where
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-- crucial facts about radical ideals
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h∈√⟨f⟩ : h ∈ √ ⟨ f ⟩[ R' ]
254-
h∈√⟨f⟩ = isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun DHelper .fst zero
254+
h∈√⟨f⟩ = √FGIdealCharLImpl _ _ (isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun DHelper .fst) zero
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f∈√⟨h⟩ : i f i ∈ √ ⟨ h ⟩ₛ
257-
f∈√⟨h⟩ i = isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun
258-
(sym DHelper) .fst i
257+
f∈√⟨h⟩ = √FGIdealCharLImpl _ _ (isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun
258+
(sym DHelper) .fst)
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260260
ff∈√⟨h⟩ : i j f i · f j ∈ √ ⟨ h ⟩ₛ
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ff∈√⟨h⟩ i j = √ ⟨ h ⟩ₛ .snd .·Closed (f i) (f∈√⟨h⟩ j)

Cubical/AlgebraicGeometry/ZariskiLattice/StructureSheafPullback.agda

Lines changed: 4 additions & 4 deletions
Original file line numberDiff line numberDiff line change
@@ -166,7 +166,7 @@ module _ (R' : CommRing ℓ) where
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Df≤Dg = subst2 _≤_ (sym p) (sym q) 𝔞≤𝔟
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f∈√⟨g⟩ : f ∈ √ ⟨ g ⟩
169-
f∈√⟨g⟩ = isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun Df≤Dg .fst zero
169+
f∈√⟨g⟩ = √FGIdealCharLImpl _ _ (isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun Df≤Dg .fst) zero
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open PreSheafFromUniversalProp ZariskiCat P 𝓕 uniqueHom
@@ -278,16 +278,16 @@ module _ (R' : CommRing ℓ) where
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DHelper = Dh≡𝔞∨𝔟 ∙ cong₂ (_∨z_) (sym Df≡𝔞) (sym Dg≡𝔟)
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f∈√⟨h⟩ : f ∈ √ ⟨ h ⟩
281-
f∈√⟨h⟩ = isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun (sym DHelper) .fst zero
281+
f∈√⟨h⟩ = √FGIdealCharLImpl _ _ (isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun (sym DHelper) .fst) zero
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g∈√⟨h⟩ : g ∈ √ ⟨ h ⟩
284-
g∈√⟨h⟩ = isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun (sym DHelper) .fst one
284+
g∈√⟨h⟩ = √FGIdealCharLImpl _ _ (isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun (sym DHelper) .fst) one
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fg∈√⟨h⟩ : (f · g) ∈ √ ⟨ h ⟩
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fg∈√⟨h⟩ = √ ⟨ h ⟩ .snd .·Closed f g∈√⟨h⟩
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1∈fgIdeal : 1r ∈ₕ ⟨ (f /1) , (g /1) ⟩ₕ
290-
1∈fgIdeal = helper1 (isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun DHelper .fst zero)
290+
1∈fgIdeal = helper1 $ √FGIdealCharLImpl _ _ (isEquivRel→effectiveIso ∼PropValued ∼EquivRel _ _ .fun DHelper .fst) zero
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where
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helper1 : h ∈ √ ⟨ f , g ⟩ₚ
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1r ∈ₕ ⟨ (f /1) , (g /1) ⟩ₕ

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