Documentation

Mathlib.AlgebraicGeometry.GammaSpecAdjunction

Adjunction between `Γ` and `Spec` #

We define the adjunction ΓSpec.adjunction : Γ ⊣ Spec by defining the unit (toΓSpec, in multiple steps in this file) and counit (done in Spec.lean) and checking that they satisfy the left and right triangle identities. The constructions and proofs make use of maps and lemmas defined and proved in structure_sheaf.lean extensively.

Notice that since the adjunction is between contravariant functors, you get to choose one of the two categories to have arrows reversed, and it is equally valid to present the adjunction as Spec ⊣ Γ (Spec.to_LocallyRingedSpace.right_op ⊣ Γ), in which case the unit and the counit would switch to each other.

Main definition #

AlgebraicGeometry.identityToΓSpec : The natural transformation 𝟭 _ ⟶ Γ ⋙ Spec.
AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction : The adjunction Γ ⊣ Spec from CommRingᵒᵖ to LocallyRingedSpace.
AlgebraicGeometry.ΓSpec.adjunction : The adjunction Γ ⊣ Spec from CommRingᵒᵖ to Scheme.

def AlgebraicGeometry.LocallyRingedSpace.ΓToStalk (X : AlgebraicGeometry.LocallyRingedSpace) (x : ↑(AlgebraicGeometry.LocallyRingedSpace.toTopCat X)) :

AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X) ⟶ TopCat.Presheaf.stalk X.presheaf x

The map from the global sections to a stalk.

Instances For

def AlgebraicGeometry.LocallyRingedSpace.toΓSpecFun (X : AlgebraicGeometry.LocallyRingedSpace) :

↑(AlgebraicGeometry.LocallyRingedSpace.toTopCat X) → PrimeSpectrum ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))

The canonical map from the underlying set to the prime spectrum of Γ(X).

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theorem AlgebraicGeometry.LocallyRingedSpace.not_mem_prime_iff_unit_in_stalk (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) (x : ↑(AlgebraicGeometry.LocallyRingedSpace.toTopCat X)) :

¬r ∈ (AlgebraicGeometry.LocallyRingedSpace.toΓSpecFun X x).asIdeal ↔ IsUnit (↑(AlgebraicGeometry.LocallyRingedSpace.ΓToStalk X x) r)

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preim_basicOpen_eq (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) :

AlgebraicGeometry.LocallyRingedSpace.toΓSpecFun X ⁻¹' (PrimeSpectrum.basicOpen r).carrier = (AlgebraicGeometry.RingedSpace.basicOpen (AlgebraicGeometry.LocallyRingedSpace.toRingedSpace X) r).carrier

The preimage of a basic open in Spec Γ(X) under the unit is the basic open in X defined by the same element (they are equal as sets).

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpec_continuous (X : AlgebraicGeometry.LocallyRingedSpace) :

Continuous (AlgebraicGeometry.LocallyRingedSpace.toΓSpecFun X)

toΓSpecFun is continuous.

@[simp]

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpecBase_apply (X : AlgebraicGeometry.LocallyRingedSpace) :

∀ (a : ↑(AlgebraicGeometry.LocallyRingedSpace.toTopCat X)), ↑(AlgebraicGeometry.LocallyRingedSpace.toΓSpecBase X) a = AlgebraicGeometry.LocallyRingedSpace.toΓSpecFun X a

def AlgebraicGeometry.LocallyRingedSpace.toΓSpecBase (X : AlgebraicGeometry.LocallyRingedSpace) :

AlgebraicGeometry.LocallyRingedSpace.toTopCat X ⟶ AlgebraicGeometry.Spec.topObj (AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))

The canonical (bundled) continuous map from the underlying topological space of X to the prime spectrum of its global sections.

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@[inline, reducible]

abbrev AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) :

TopologicalSpace.Opens ↑(AlgebraicGeometry.LocallyRingedSpace.toTopCat X)

The preimage in X of a basic open in Spec Γ(X) (as an open set).

Instances For

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen_eq (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) :

AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen X r = AlgebraicGeometry.RingedSpace.basicOpen (AlgebraicGeometry.LocallyRingedSpace.toRingedSpace X) r

The preimage is the basic open in X defined by the same element r.

@[inline, reducible]

abbrev AlgebraicGeometry.LocallyRingedSpace.toToΓSpecMapBasicOpen (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) :

X.presheaf.obj (Opposite.op ⊤) ⟶ X.presheaf.obj (Opposite.op (AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen X r))

The map from the global sections Γ(X) to the sections on the (preimage of) a basic open.

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theorem AlgebraicGeometry.LocallyRingedSpace.isUnit_res_toΓSpecMapBasicOpen (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) :

IsUnit (↑(AlgebraicGeometry.LocallyRingedSpace.toToΓSpecMapBasicOpen X r) r)

r is a unit as a section on the basic open defined by r.

def AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) :

(AlgebraicGeometry.Spec.structureSheaf ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))).val.obj (Opposite.op (PrimeSpectrum.basicOpen r)) ⟶ X.presheaf.obj (Opposite.op (AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen X r))

Define the sheaf hom on individual basic opens for the unit.

Instances For

Characterization of the sheaf hom on basic opens, direction ← (next lemma) is used at various places, but → is not used in this file.

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp_spec (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) :

CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.StructureSheaf.toOpen (↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) (PrimeSpectrum.basicOpen r)) (AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp X r) = AlgebraicGeometry.LocallyRingedSpace.toToΓSpecMapBasicOpen X r

@[simp]

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens_app (X : AlgebraicGeometry.LocallyRingedSpace) (r : (CategoryTheory.InducedCategory (TopologicalSpace.Opens (PrimeSpectrum ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X)))) PrimeSpectrum.basicOpen)ᵒᵖ) :

(AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens X).app r = AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp X r.unop

def AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens (X : AlgebraicGeometry.LocallyRingedSpace) :

CategoryTheory.Functor.comp (CategoryTheory.inducedFunctor PrimeSpectrum.basicOpen).op (AlgebraicGeometry.Spec.structureSheaf ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))).val ⟶ CategoryTheory.Functor.comp (CategoryTheory.inducedFunctor PrimeSpectrum.basicOpen).op ((TopCat.Sheaf.pushforward (AlgebraicGeometry.LocallyRingedSpace.toΓSpecBase X)).obj (AlgebraicGeometry.LocallyRingedSpace.𝒪 X)).val

The sheaf hom on all basic opens, commuting with restrictions.

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@[simp]

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_base (X : AlgebraicGeometry.LocallyRingedSpace) :

(AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace X).base = AlgebraicGeometry.LocallyRingedSpace.toΓSpecBase X

@[simp]

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_c (X : AlgebraicGeometry.LocallyRingedSpace) :

(AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace X).c = ↑(TopCat.Sheaf.restrictHomEquivHom (AlgebraicGeometry.Spec.structureSheaf ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))).val ((TopCat.Sheaf.pushforward (AlgebraicGeometry.LocallyRingedSpace.toΓSpecBase X)).obj (AlgebraicGeometry.LocallyRingedSpace.𝒪 X)) (_ : TopologicalSpace.Opens.IsBasis (Set.range PrimeSpectrum.basicOpen))) (AlgebraicGeometry.LocallyRingedSpace.toΓSpecCBasicOpens X)

def AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace (X : AlgebraicGeometry.LocallyRingedSpace) :

X.toSheafedSpace ⟶ AlgebraicGeometry.Spec.toSheafedSpace.obj (Opposite.op (AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X)))

The canonical morphism of sheafed spaces from X to the spectrum of its global sections.

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theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_eq (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) :

(AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace X).c.app (Opposite.op (PrimeSpectrum.basicOpen r)) = AlgebraicGeometry.LocallyRingedSpace.toΓSpecCApp X r

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_app_spec (X : AlgebraicGeometry.LocallyRingedSpace) (r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) :

CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.StructureSheaf.toOpen (↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) (PrimeSpectrum.basicOpen r)) ((AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace X).c.app (Opposite.op (PrimeSpectrum.basicOpen r))) = AlgebraicGeometry.LocallyRingedSpace.toToΓSpecMapBasicOpen X r

theorem AlgebraicGeometry.LocallyRingedSpace.toStalk_stalkMap_toΓSpec (X : AlgebraicGeometry.LocallyRingedSpace) (x : ↑(AlgebraicGeometry.LocallyRingedSpace.toTopCat X)) :

CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.StructureSheaf.toStalk (↑(Opposite.op (AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))).unop) (↑(AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace X).base x)) (AlgebraicGeometry.PresheafedSpace.stalkMap (AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace X) x) = AlgebraicGeometry.LocallyRingedSpace.ΓToStalk X x

The map on stalks induced by the unit commutes with maps from Γ(X) to stalks (in Spec Γ(X) and in X).

@[simp]

theorem AlgebraicGeometry.LocallyRingedSpace.toΓSpec_val_base (X : AlgebraicGeometry.LocallyRingedSpace) :

(AlgebraicGeometry.LocallyRingedSpace.toΓSpec X).val.base = AlgebraicGeometry.LocallyRingedSpace.toΓSpecBase X

def AlgebraicGeometry.LocallyRingedSpace.toΓSpec (X : AlgebraicGeometry.LocallyRingedSpace) :

X ⟶ AlgebraicGeometry.Spec.locallyRingedSpaceObj (AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))

The canonical morphism from X to the spectrum of its global sections.

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theorem AlgebraicGeometry.LocallyRingedSpace.comp_ring_hom_ext {X : AlgebraicGeometry.LocallyRingedSpace} {R : CommRingCat} {f : R ⟶ AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X)} {β : X ⟶ AlgebraicGeometry.Spec.locallyRingedSpaceObj R} (w : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.LocallyRingedSpace.toΓSpec X).val.base (AlgebraicGeometry.Spec.locallyRingedSpaceMap f).val.base = β.val.base) (h : ∀ (r : ↑R), CategoryTheory.CategoryStruct.comp f (X.presheaf.map (CategoryTheory.homOfLE (_ : (TopologicalSpace.Opens.map β.val.base).obj (PrimeSpectrum.basicOpen r) ≤ ⊤)).op) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.StructureSheaf.toOpen (↑R) (PrimeSpectrum.basicOpen r)) (β.val.c.app (Opposite.op (PrimeSpectrum.basicOpen r)))) :

CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.LocallyRingedSpace.toΓSpec X) (AlgebraicGeometry.Spec.locallyRingedSpaceMap f) = β

theorem AlgebraicGeometry.LocallyRingedSpace.Γ_Spec_left_triangle (X : AlgebraicGeometry.LocallyRingedSpace) :

CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.toSpecΓ (AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) ((AlgebraicGeometry.LocallyRingedSpace.toΓSpec X).val.c.app (Opposite.op ⊤)) = CategoryTheory.CategoryStruct.id (AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))

toSpecΓ _ is an isomorphism so these are mutually two-sided inverses.

def AlgebraicGeometry.identityToΓSpec :

CategoryTheory.Functor.id AlgebraicGeometry.LocallyRingedSpace ⟶ CategoryTheory.Functor.comp AlgebraicGeometry.LocallyRingedSpace.Γ.rightOp AlgebraicGeometry.Spec.toLocallyRingedSpace

The unit as a natural transformation.

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theorem AlgebraicGeometry.ΓSpec.left_triangle (X : AlgebraicGeometry.LocallyRingedSpace) :

CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.SpecΓIdentity.inv.app (AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))) ((AlgebraicGeometry.identityToΓSpec.app X).val.c.app (Opposite.op ⊤)) = CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.id CommRingCat).obj (AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X)))

theorem AlgebraicGeometry.ΓSpec.right_triangle (R : CommRingCat) :

CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.identityToΓSpec.app (AlgebraicGeometry.Spec.toLocallyRingedSpace.obj (Opposite.op R))) (AlgebraicGeometry.Spec.toLocallyRingedSpace.map (AlgebraicGeometry.SpecΓIdentity.inv.app R).op) = CategoryTheory.CategoryStruct.id ((CategoryTheory.Functor.id AlgebraicGeometry.LocallyRingedSpace).obj (AlgebraicGeometry.Spec.toLocallyRingedSpace.obj (Opposite.op R)))

SpecΓIdentity is iso so these are mutually two-sided inverses.

def AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction :

AlgebraicGeometry.LocallyRingedSpace.Γ.rightOp ⊣ AlgebraicGeometry.Spec.toLocallyRingedSpace

The adjunction Γ ⊣ Spec from CommRingᵒᵖ to LocallyRingedSpace.

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theorem AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_unit :

AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction.unit = AlgebraicGeometry.identityToΓSpec

theorem AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit :

AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction.counit = (CategoryTheory.NatIso.op AlgebraicGeometry.SpecΓIdentity).inv

def AlgebraicGeometry.ΓSpec.adjunction :

AlgebraicGeometry.Scheme.Γ.rightOp ⊣ AlgebraicGeometry.Scheme.Spec

The adjunction Γ ⊣ Spec from CommRingᵒᵖ to Scheme.

Instances For

theorem AlgebraicGeometry.ΓSpec.adjunction_homEquiv_apply {X : AlgebraicGeometry.Scheme} {R : CommRingCat ᵒᵖ} (f : Opposite.op (AlgebraicGeometry.Scheme.Γ.obj (Opposite.op X)) ⟶ R) :

↑(CategoryTheory.Adjunction.homEquiv AlgebraicGeometry.ΓSpec.adjunction X R) f = ↑(CategoryTheory.Adjunction.homEquiv AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction X.toLocallyRingedSpace R) f

theorem AlgebraicGeometry.ΓSpec.adjunction_homEquiv (X : AlgebraicGeometry.Scheme) (R : CommRingCat ᵒᵖ) :

CategoryTheory.Adjunction.homEquiv AlgebraicGeometry.ΓSpec.adjunction X R = CategoryTheory.Adjunction.homEquiv AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction X.toLocallyRingedSpace R

theorem AlgebraicGeometry.ΓSpec.adjunction_homEquiv_symm_apply {X : AlgebraicGeometry.Scheme} {R : CommRingCat ᵒᵖ} (f : X ⟶ AlgebraicGeometry.Scheme.Spec.obj R) :

↑(CategoryTheory.Adjunction.homEquiv AlgebraicGeometry.ΓSpec.adjunction X R).symm f = ↑(CategoryTheory.Adjunction.homEquiv AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction X.toLocallyRingedSpace R).symm f

@[simp]

theorem AlgebraicGeometry.ΓSpec.adjunction_counit_app {R : CommRingCat ᵒᵖ} :

AlgebraicGeometry.ΓSpec.adjunction.counit.app R = AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction.counit.app R

@[simp]

theorem AlgebraicGeometry.ΓSpec.adjunction_unit_app {X : AlgebraicGeometry.Scheme} :

AlgebraicGeometry.ΓSpec.adjunction.unit.app X = AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction.unit.app X.toLocallyRingedSpace

instance AlgebraicGeometry.ΓSpec.isIso_locallyRingedSpaceAdjunction_counit :

CategoryTheory.IsIso AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction.counit

instance AlgebraicGeometry.ΓSpec.isIso_adjunction_counit :

CategoryTheory.IsIso AlgebraicGeometry.ΓSpec.adjunction.counit

theorem AlgebraicGeometry.ΓSpec.adjunction_unit_app_app_top (X : AlgebraicGeometry.Scheme) :

(AlgebraicGeometry.ΓSpec.adjunction.unit.app X).val.c.app (Opposite.op ⊤) = AlgebraicGeometry.SpecΓIdentity.hom.app (X.presheaf.obj (Opposite.op ⊤))

Immediate consequences of the adjunction.

instance AlgebraicGeometry.instPreservesLimitsOppositeCommRingCatOppositeInstCommRingCatLargeCategoryLocallyRingedSpaceInstCategoryLocallyRingedSpaceToLocallyRingedSpace :

CategoryTheory.Limits.PreservesLimits AlgebraicGeometry.Spec.toLocallyRingedSpace

Spec preserves limits.

instance AlgebraicGeometry.Spec.preservesLimits :

CategoryTheory.Limits.PreservesLimits AlgebraicGeometry.Scheme.Spec

instance AlgebraicGeometry.instFullOppositeCommRingCatOppositeInstCommRingCatLargeCategoryLocallyRingedSpaceInstCategoryLocallyRingedSpaceToLocallyRingedSpace :

CategoryTheory.Full AlgebraicGeometry.Spec.toLocallyRingedSpace

Spec is a full functor.

instance AlgebraicGeometry.Spec.full :

CategoryTheory.Full AlgebraicGeometry.Scheme.Spec

instance AlgebraicGeometry.instFaithfulOppositeCommRingCatOppositeInstCommRingCatLargeCategoryLocallyRingedSpaceInstCategoryLocallyRingedSpaceToLocallyRingedSpace :

CategoryTheory.Faithful AlgebraicGeometry.Spec.toLocallyRingedSpace

Spec is a faithful functor.

instance AlgebraicGeometry.Spec.faithful :

CategoryTheory.Faithful AlgebraicGeometry.Scheme.Spec

instance AlgebraicGeometry.instIsRightAdjointLocallyRingedSpaceInstCategoryLocallyRingedSpaceOppositeCommRingCatOppositeInstCommRingCatLargeCategoryToLocallyRingedSpace :

CategoryTheory.IsRightAdjoint AlgebraicGeometry.Spec.toLocallyRingedSpace

instance AlgebraicGeometry.instIsRightAdjointSchemeInstCategorySchemeOppositeCommRingCatOppositeInstCommRingCatLargeCategorySpec :

CategoryTheory.IsRightAdjoint AlgebraicGeometry.Scheme.Spec

instance AlgebraicGeometry.instReflectiveLocallyRingedSpaceOppositeCommRingCatInstCategoryLocallyRingedSpaceOppositeInstCommRingCatLargeCategoryToLocallyRingedSpace :

CategoryTheory.Reflective AlgebraicGeometry.Spec.toLocallyRingedSpace

instance AlgebraicGeometry.Spec.reflective :

CategoryTheory.Reflective AlgebraicGeometry.Scheme.Spec