fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
toStalk_comp_stalkToFiberRingHom (x : PrimeSpectrum.Top R) :
toStalk R x ≫ stalkToFiberRingHom R x = CommRingCat.ofHom (algebraMap _ _) := by
rw [toStalk, Category.assoc, germ_comp_stalkToFiberRingHom]; rfl
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toStalk_comp_stalkToFiberRingHom | null |
stalkToFiberRingHom_toStalk (x : PrimeSpectrum.Top R) (f : R) :
stalkToFiberRingHom R x (toStalk R x f) = algebraMap _ _ f :=
RingHom.ext_iff.1 (CommRingCat.hom_ext_iff.mp (toStalk_comp_stalkToFiberRingHom R x)) _ | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | stalkToFiberRingHom_toStalk | null |
@[simps]
stalkIso (x : PrimeSpectrum.Top R) :
(structureSheaf R).presheaf.stalk x ≅ CommRingCat.of (Localization.AtPrime x.asIdeal) where
hom := stalkToFiberRingHom R x
inv := localizationToStalk R x
hom_inv_id := by
apply stalk_hom_ext
intro U hxU
ext s
dsimp only [CommRingCat.hom_comp, RingH... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | stalkIso | The ring isomorphism between the stalk of the structure sheaf of `R` at a point `p`
corresponding to a prime ideal in `R` and the localization of `R` at `p`. |
@[simp, reassoc]
stalkToFiberRingHom_localizationToStalk (x : PrimeSpectrum.Top R) :
stalkToFiberRingHom R x ≫ localizationToStalk R x = 𝟙 _ :=
(stalkIso R x).hom_inv_id
@[simp, reassoc] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | stalkToFiberRingHom_localizationToStalk | null |
localizationToStalk_stalkToFiberRingHom (x : PrimeSpectrum.Top R) :
localizationToStalk R x ≫ stalkToFiberRingHom R x = 𝟙 _ :=
(stalkIso R x).inv_hom_id | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | localizationToStalk_stalkToFiberRingHom | null |
toBasicOpen (f : R) :
Localization.Away f →+* (structureSheaf R).1.obj (op <| PrimeSpectrum.basicOpen f) :=
IsLocalization.Away.lift f (isUnit_to_basicOpen_self R f)
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toBasicOpen | The canonical ring homomorphism interpreting `s ∈ R_f` as a section of the structure sheaf
on the basic open defined by `f ∈ R`. |
toBasicOpen_mk' (s f : R) (g : Submonoid.powers s) :
toBasicOpen R s (IsLocalization.mk' (Localization.Away s) f g) =
const R f g (PrimeSpectrum.basicOpen s) fun _ hx => Submonoid.powers_le.2 hx g.2 :=
(IsLocalization.lift_mk'_spec _ _ _ _).2 <| by
rw [toOpen_eq_const, toOpen_eq_const, const_mul_cancel'... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toBasicOpen_mk' | null |
localization_toBasicOpen (f : R) :
RingHom.comp (toBasicOpen R f) (algebraMap R (Localization.Away f)) =
(toOpen R (PrimeSpectrum.basicOpen f)).hom :=
RingHom.ext fun g => by
rw [toBasicOpen, IsLocalization.Away.lift, RingHom.comp_apply, IsLocalization.lift_eq]
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | localization_toBasicOpen | null |
toBasicOpen_to_map (s f : R) :
toBasicOpen R s (algebraMap R (Localization.Away s) f) =
const R f 1 (PrimeSpectrum.basicOpen s) fun _ _ => Submonoid.one_mem _ :=
(IsLocalization.lift_eq _ _).trans <| toOpen_eq_const _ _ _ | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toBasicOpen_to_map | null |
toBasicOpen_injective (f : R) : Function.Injective (toBasicOpen R f) := by
intro s t h_eq
obtain ⟨a, ⟨b, hb⟩, rfl⟩ := IsLocalization.mk'_surjective (Submonoid.powers f) s
obtain ⟨c, ⟨d, hd⟩, rfl⟩ := IsLocalization.mk'_surjective (Submonoid.powers f) t
simp only [toBasicOpen_mk'] at h_eq
rw [IsLocalization.eq]... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toBasicOpen_injective | null |
locally_const_basicOpen (U : Opens (PrimeSpectrum.Top R))
(s : (structureSheaf R).1.obj (op U)) (x : U) :
∃ (f g : R) (i : PrimeSpectrum.basicOpen g ⟶ U), x.1 ∈ PrimeSpectrum.basicOpen g ∧
(const R f g (PrimeSpectrum.basicOpen g) fun _ hy => hy) =
(structureSheaf R).1.map i.op s := by
obtain ⟨V, h... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | locally_const_basicOpen | null |
normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
(s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
(iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
(h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
(hs :
∀ i : ι,
(const R (a i) ... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | normalize_finite_fraction_representation | null |
toBasicOpen_surjective (f : R) : Function.Surjective (toBasicOpen R f) := by
intro s
let ι : Type u := PrimeSpectrum.basicOpen f
choose a' h' iDh' hxDh' s_eq' using locally_const_basicOpen R (PrimeSpectrum.basicOpen f) s
obtain ⟨t, ht_cover'⟩ :=
(PrimeSpectrum.isCompact_basicOpen f).elim_finite_subcover
... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toBasicOpen_surjective | null |
isIso_toBasicOpen (f : R) :
IsIso (CommRingCat.ofHom (toBasicOpen R f)) :=
haveI : IsIso ((forget CommRingCat).map (CommRingCat.ofHom (toBasicOpen R f))) :=
(isIso_iff_bijective _).mpr ⟨toBasicOpen_injective R f, toBasicOpen_surjective R f⟩
isIso_of_reflects_iso _ (forget CommRingCat) | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | isIso_toBasicOpen | null |
basicOpenIso (f : R) :
(structureSheaf R).1.obj (op (PrimeSpectrum.basicOpen f)) ≅
CommRingCat.of (Localization.Away f) :=
(asIso (CommRingCat.ofHom (toBasicOpen R f))).symm | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | basicOpenIso | The ring isomorphism between the structure sheaf on `basicOpen f` and the localization of `R`
at the submonoid of powers of `f`. |
stalkAlgebra (p : PrimeSpectrum R) : Algebra R ((structureSheaf R).presheaf.stalk p) :=
(toStalk R p).hom.toAlgebra
@[simp] | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | stalkAlgebra | null |
stalkAlgebra_map (p : PrimeSpectrum R) (r : R) :
algebraMap R ((structureSheaf R).presheaf.stalk p) r = toStalk R p r :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | stalkAlgebra_map | null |
IsLocalization.to_stalk (p : PrimeSpectrum R) :
IsLocalization.AtPrime ((structureSheaf R).presheaf.stalk p) p.asIdeal := by
convert (IsLocalization.isLocalization_iff_of_ringEquiv (S := Localization.AtPrime p.asIdeal) _
(stalkIso R p).symm.commRingCatIsoToRingEquiv).mp
Localization.isLocalization... | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | IsLocalization.to_stalk | Stalk of the structure sheaf at a prime p as localization of R |
openAlgebra (U : (Opens (PrimeSpectrum R))ᵒᵖ) : Algebra R ((structureSheaf R).val.obj U) :=
(toOpen R (unop U)).hom.toAlgebra
@[simp] | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | openAlgebra | null |
openAlgebra_map (U : (Opens (PrimeSpectrum R))ᵒᵖ) (r : R) :
algebraMap R ((structureSheaf R).val.obj U) r = toOpen R (unop U) r :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | openAlgebra_map | null |
IsLocalization.to_basicOpen (r : R) :
IsLocalization.Away r ((structureSheaf R).val.obj (op <| PrimeSpectrum.basicOpen r)) := by
convert (IsLocalization.isLocalization_iff_of_ringEquiv (S := Localization.Away r) _
(basicOpenIso R r).symm.commRingCatIsoToRingEquiv).mp
Localization.isLocalization
appl... | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | IsLocalization.to_basicOpen | Sections of the structure sheaf of Spec R on a basic open as localization of R |
to_basicOpen_epi (r : R) : Epi (toOpen R (PrimeSpectrum.basicOpen r)) :=
⟨fun _ _ h => CommRingCat.hom_ext (IsLocalization.ringHom_ext (Submonoid.powers r)
(CommRingCat.hom_ext_iff.mp h))⟩
@[elementwise] | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | to_basicOpen_epi | null |
to_global_factors :
toOpen R ⊤ =
CommRingCat.ofHom (algebraMap R (Localization.Away (1 : R))) ≫
CommRingCat.ofHom (toBasicOpen R (1 : R)) ≫
(structureSheaf R).1.map (eqToHom PrimeSpectrum.basicOpen_one.symm).op := by
rw [← Category.assoc]
change toOpen R ⊤ =
(CommRingCat.ofHom <| (toBa... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | to_global_factors | null |
isIso_to_global : IsIso (toOpen R ⊤) := by
let hom := CommRingCat.ofHom (algebraMap R (Localization.Away (1 : R)))
haveI : IsIso hom :=
(IsLocalization.atOne R (Localization.Away (1 : R))).toRingEquiv.toCommRingCatIso.isIso_hom
rw [to_global_factors R]
infer_instance | instance | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | isIso_to_global | null |
@[simps! inv]
globalSectionsIso : CommRingCat.of R ≅ (structureSheaf R).1.obj (op ⊤) :=
asIso (toOpen R ⊤)
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | globalSectionsIso | The ring isomorphism between the ring `R` and the global sections `Γ(X, 𝒪ₓ)`. |
globalSectionsIso_hom (R : CommRingCat) : (globalSectionsIso R).hom = toOpen R ⊤ :=
rfl
@[simp, reassoc, elementwise nosimp] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | globalSectionsIso_hom | null |
toStalk_stalkSpecializes {R : Type*} [CommRing R] {x y : PrimeSpectrum R} (h : x ⤳ y) :
toStalk R y ≫ (structureSheaf R).presheaf.stalkSpecializes h = toStalk R x := by
dsimp [toStalk]; simp [-toOpen_germ]
@[simp, reassoc, elementwise nosimp] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toStalk_stalkSpecializes | null |
localizationToStalk_stalkSpecializes {R : Type*} [CommRing R] {x y : PrimeSpectrum R}
(h : x ⤳ y) :
StructureSheaf.localizationToStalk R y ≫ (structureSheaf R).presheaf.stalkSpecializes h =
CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h) ≫
StructureSheaf.localizationToStalk R x :=... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | localizationToStalk_stalkSpecializes | null |
stalkSpecializes_stalk_to_fiber {R : Type*} [CommRing R] {x y : PrimeSpectrum R}
(h : x ⤳ y) :
(structureSheaf R).presheaf.stalkSpecializes h ≫ StructureSheaf.stalkToFiberRingHom R x =
StructureSheaf.stalkToFiberRingHom R y ≫
(CommRingCat.ofHom (PrimeSpectrum.localizationMapOfSpecializes h)) := by
... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | stalkSpecializes_stalk_to_fiber | null |
comapFun (f : R →+* S) (U : Opens (PrimeSpectrum.Top R)) (V : Opens (PrimeSpectrum.Top S))
(hUV : V.1 ⊆ PrimeSpectrum.comap f ⁻¹' U.1) (s : ∀ x : U, Localizations R x) (y : V) :
Localizations S y :=
Localization.localRingHom (PrimeSpectrum.comap f y.1).asIdeal _ f rfl
(s ⟨PrimeSpectrum.comap f y.1, hUV y.... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comapFun | Given a ring homomorphism `f : R →+* S`, an open set `U` of the prime spectrum of `R` and an open
set `V` of the prime spectrum of `S`, such that `V ⊆ (comap f) ⁻¹' U`, we can push a section `s`
on `U` to a section on `V`, by composing with `Localization.localRingHom _ _ f` from the left and
`comap f` from the right. E... |
comapFunIsLocallyFraction (f : R →+* S) (U : Opens (PrimeSpectrum.Top R))
(V : Opens (PrimeSpectrum.Top S)) (hUV : V.1 ⊆ PrimeSpectrum.comap f ⁻¹' U.1)
(s : ∀ x : U, Localizations R x) (hs : (isLocallyFraction R).toPrelocalPredicate.pred s) :
(isLocallyFraction S).toPrelocalPredicate.pred (comapFun f U V hU... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comapFunIsLocallyFraction | null |
comap (f : R →+* S) (U : Opens (PrimeSpectrum.Top R)) (V : Opens (PrimeSpectrum.Top S))
(hUV : V.1 ⊆ PrimeSpectrum.comap f ⁻¹' U.1) :
(structureSheaf R).1.obj (op U) →+* (structureSheaf S).1.obj (op V) where
toFun s := ⟨comapFun f U V hUV s.1, comapFunIsLocallyFraction f U V hUV s.1 s.2⟩
map_one' :=
Sub... | def | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comap | For a ring homomorphism `f : R →+* S` and open sets `U` and `V` of the prime spectra of `R` and
`S` such that `V ⊆ (comap f) ⁻¹ U`, the induced ring homomorphism from the structure sheaf of `R`
at `U` to the structure sheaf of `S` at `V`.
Explicitly, this map is given as follows: For a point `p : V`, if the section `s... |
comap_apply (f : R →+* S) (U : Opens (PrimeSpectrum.Top R))
(V : Opens (PrimeSpectrum.Top S)) (hUV : V.1 ⊆ PrimeSpectrum.comap f ⁻¹' U.1)
(s : (structureSheaf R).1.obj (op U)) (p : V) :
(comap f U V hUV s).1 p =
Localization.localRingHom (PrimeSpectrum.comap f p.1).asIdeal _ f rfl
(s.1 ⟨PrimeS... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comap_apply | null |
comap_const (f : R →+* S) (U : Opens (PrimeSpectrum.Top R))
(V : Opens (PrimeSpectrum.Top S)) (hUV : V.1 ⊆ PrimeSpectrum.comap f ⁻¹' U.1) (a b : R)
(hb : ∀ x : PrimeSpectrum R, x ∈ U → b ∈ x.asIdeal.primeCompl) :
comap f U V hUV (const R a b U hb) =
const S (f a) (f b) V fun p hpV => hb (PrimeSpectrum... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comap_const | null |
comap_id_eq_map (U V : Opens (PrimeSpectrum.Top R)) (iVU : V ⟶ U) :
(comap (RingHom.id R) U V fun _ hpV => leOfHom iVU <| hpV) =
((structureSheaf R).1.map iVU.op).hom :=
RingHom.ext fun s => Subtype.eq <| funext fun p => by
rw [comap_apply]
obtain ⟨W, hpW, iWU, h⟩ := s.2 (iVU p)
obtain ⟨a, b, h'... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comap_id_eq_map | For an inclusion `i : V ⟶ U` between open sets of the prime spectrum of `R`, the comap of the
identity from OO_X(U) to OO_X(V) equals as the restriction map of the structure sheaf.
This is a generalization of the fact that, for fixed `U`, the comap of the identity from OO_X(U)
to OO_X(U) is the identity. |
comap_id {U V : Opens (PrimeSpectrum.Top R)} (hUV : U = V) :
(comap (RingHom.id R) U V fun p hpV => by rwa [hUV, PrimeSpectrum.comap_id]) =
(eqToHom (show (structureSheaf R).1.obj (op U) = _ by rw [hUV])).hom := by
rw [comap_id_eq_map U V (eqToHom hUV.symm), eqToHom_op, eqToHom_map]
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comap_id | The comap of the identity is the identity. In this variant of the lemma, two open subsets `U` and
`V` are given as arguments, together with a proof that `U = V`. This is useful when `U` and `V`
are not definitionally equal. |
comap_id' (U : Opens (PrimeSpectrum.Top R)) :
(comap (RingHom.id R) U U fun p hpU => by rwa [PrimeSpectrum.comap_id]) = RingHom.id _ := by
rw [comap_id rfl]; rfl | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comap_id' | null |
comap_comp (f : R →+* S) (g : S →+* P) (U : Opens (PrimeSpectrum.Top R))
(V : Opens (PrimeSpectrum.Top S)) (W : Opens (PrimeSpectrum.Top P))
(hUV : ∀ p ∈ V, PrimeSpectrum.comap f p ∈ U) (hVW : ∀ p ∈ W, PrimeSpectrum.comap g p ∈ V) :
(comap (g.comp f) U W fun p hpW => hUV (PrimeSpectrum.comap g p) (hVW p hpW... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comap_comp | null |
toOpen_comp_comap (f : R →+* S) (U : Opens (PrimeSpectrum.Top R)) :
(toOpen R U ≫ CommRingCat.ofHom (comap f U (Opens.comap (PrimeSpectrum.comap f) U)
fun _ => id)) =
CommRingCat.ofHom f ≫ toOpen S _ :=
CommRingCat.hom_ext <| RingHom.ext fun _ => Subtype.eq <| funext fun _ =>
Localization.localR... | theorem | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | toOpen_comp_comap | null |
comap_basicOpen (f : R →+* S) (x : R) :
comap f (PrimeSpectrum.basicOpen x) (PrimeSpectrum.basicOpen (f x))
(PrimeSpectrum.comap_basicOpen f x).le =
IsLocalization.map (M := .powers x) (T := .powers (f x)) _ f
(Submonoid.powers_le.mpr (Submonoid.mem_powers _)) :=
IsLocalization.ringHom_ext (... | lemma | AlgebraicGeometry | [
"Mathlib.Algebra.Category.Ring.Colimits",
"Mathlib.Algebra.Category.Ring.Instances",
"Mathlib.Algebra.Category.Ring.Limits",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.RingTheory.Localization.AtPrime.Basic",
"Mathlib.RingTheory.Spectrum.Prime.Topology",
"Mathlib.Topology.Sheaves.LocalPredicate"
] | Mathlib/AlgebraicGeometry/StructureSheaf.lean | comap_basicOpen | null |
ValuativeCommSq {X Y : Scheme.{u}} (f : X ⟶ Y) where
/-- The valuation ring of a valuative commutative square. -/
R : Type u
[commRing : CommRing R]
[domain : IsDomain R]
[valuationRing : ValuationRing R]
/-- The field of fractions of a valuative commutative square. -/
K : Type u
[field : Field K]
[al... | structure | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | ValuativeCommSq | A valuative commutative square over a morphism `f : X ⟶ Y` is a square
```
Spec K ⟶ Y
| |
↓ ↓
Spec R ⟶ X
```
where `R` is a valuation ring, and `K` is its ring of fractions.
We are interested in finding lifts `Spec R ⟶ Y` of this diagram. |
ValuativeCriterion.Existence : MorphismProperty Scheme :=
fun _ _ f ↦ ∀ S : ValuativeCommSq f, S.commSq.HasLift | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | ValuativeCriterion.Existence | A morphism `f : X ⟶ Y` satisfies the existence part of the valuative criterion if
every valuative commutative square over `f` has (at least) a lift. |
ValuativeCriterion.Uniqueness : MorphismProperty Scheme :=
fun _ _ f ↦ ∀ S : ValuativeCommSq f, Subsingleton S.commSq.LiftStruct | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | ValuativeCriterion.Uniqueness | A morphism `f : X ⟶ Y` satisfies the uniqueness part of the valuative criterion if
every valuative commutative square over `f` has at most one lift. |
ValuativeCriterion : MorphismProperty Scheme :=
fun _ _ f ↦ ∀ S : ValuativeCommSq f, Nonempty (Unique (S.commSq.LiftStruct))
variable {X Y : Scheme.{u}} (f : X ⟶ Y) | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | ValuativeCriterion | A morphism `f : X ⟶ Y` satisfies the valuative criterion if
every valuative commutative square over `f` has a unique lift. |
ValuativeCriterion.iff {f : X ⟶ Y} :
ValuativeCriterion f ↔ Existence f ∧ Uniqueness f := by
change (∀ _, _) ↔ (∀ _, _) ∧ (∀ _, _)
simp_rw [← forall_and, unique_iff_subsingleton_and_nonempty, and_comm, CommSq.HasLift.iff] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | ValuativeCriterion.iff | null |
ValuativeCriterion.eq :
ValuativeCriterion = Existence ⊓ Uniqueness := by
ext X Y f
exact iff | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | ValuativeCriterion.eq | null |
ValuativeCriterion.existence {f : X ⟶ Y} (h : ValuativeCriterion f) :
ValuativeCriterion.Existence f := (iff.mp h).1 | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | ValuativeCriterion.existence | null |
ValuativeCriterion.uniqueness {f : X ⟶ Y} (h : ValuativeCriterion f) :
ValuativeCriterion.Uniqueness f := (iff.mp h).2 | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | ValuativeCriterion.uniqueness | null |
@[stacks 01KE]
specializingMap (H : ValuativeCriterion.Existence f) :
SpecializingMap f.base := by
intro x' y h
let stalk_y_to_residue_x' : Y.presheaf.stalk y ⟶ X.residueField x' :=
Y.presheaf.stalkSpecializes h ≫ f.stalkMap x' ≫ X.residue x'
obtain ⟨A, hA, hA_local⟩ := exists_factor_valuationRing stalk_y... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | specializingMap | null |
of_specializingMap (H : (topologically @SpecializingMap).universally f) :
ValuativeCriterion.Existence f := by
rintro ⟨R, K, i₁, i₂, ⟨w⟩⟩
haveI : IsDomain (CommRingCat.of R) := ‹_›
haveI : ValuationRing (CommRingCat.of R) := ‹_›
letI : Field (CommRingCat.of K) := ‹_›
replace H := H (pullback.snd i₂ f) i₂ ... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | of_specializingMap | null |
stableUnderBaseChange : ValuativeCriterion.Existence.IsStableUnderBaseChange := by
constructor
intro Y' X X' Y Y'_to_Y f X'_to_X f' hP hf commSq
let commSq' : ValuativeCommSq f :=
{ R := commSq.R
K := commSq.K
i₁ := commSq.i₁ ≫ X'_to_X
i₂ := commSq.i₂ ≫ Y'_to_Y
commSq := ⟨by simp only [Category... | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | stableUnderBaseChange | null |
protected eq :
ValuativeCriterion.Existence = (topologically @SpecializingMap).universally := by
ext
constructor
· intro _
apply MorphismProperty.universally_mono
· apply specializingMap
· rwa [MorphismProperty.IsStableUnderBaseChange.universally_eq]
· apply of_specializingMap | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | eq | null |
@[stacks 01KF]
UniversallyClosed.eq_valuativeCriterion :
@UniversallyClosed = ValuativeCriterion.Existence ⊓ @QuasiCompact := by
rw [universallyClosed_eq_universallySpecializing, ValuativeCriterion.Existence.eq] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | UniversallyClosed.eq_valuativeCriterion | The **valuative criterion** for universally closed morphisms. |
@[stacks 01KF]
UniversallyClosed.of_valuativeCriterion [QuasiCompact f]
(hf : ValuativeCriterion.Existence f) : UniversallyClosed f := by
rw [eq_valuativeCriterion]
exact ⟨hf, ‹_›⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | UniversallyClosed.of_valuativeCriterion | The **valuative criterion** for universally closed morphisms. |
@[stacks 01L0]
IsSeparated.of_valuativeCriterion [QuasiSeparated f]
(hf : ValuativeCriterion.Uniqueness f) : IsSeparated f where
diagonal_isClosedImmersion := by
suffices h : ValuativeCriterion.Existence (pullback.diagonal f) by
have : QuasiCompact (pullback.diagonal f) :=
AlgebraicGeometry.Quas... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | IsSeparated.of_valuativeCriterion | The **valuative criterion** for separated morphisms. |
IsSeparated.valuativeCriterion [IsSeparated f] : ValuativeCriterion.Uniqueness f := by
intro S
constructor
rintro ⟨l₁, hl₁, hl₁'⟩ ⟨l₂, hl₂, hl₂'⟩
ext : 1
dsimp at *
have h := hl₁'.trans hl₂'.symm
let Z := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
let g : Z ⟶ Spec(S.R) := pullback.snd _ _
... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | IsSeparated.valuativeCriterion | null |
IsSeparated.eq_valuativeCriterion :
@IsSeparated = ValuativeCriterion.Uniqueness ⊓ @QuasiSeparated := by
ext X Y f
exact ⟨fun _ ↦ ⟨IsSeparated.valuativeCriterion f, inferInstance⟩,
fun ⟨H, _⟩ ↦ .of_valuativeCriterion f H⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | IsSeparated.eq_valuativeCriterion | The **valuative criterion** for separated morphisms. |
@[stacks 0BX5]
IsProper.eq_valuativeCriterion :
@IsProper = ValuativeCriterion ⊓ @QuasiCompact ⊓ @QuasiSeparated ⊓ @LocallyOfFiniteType := by
rw [isProper_eq, IsSeparated.eq_valuativeCriterion, ValuativeCriterion.eq,
UniversallyClosed.eq_valuativeCriterion]
simp_rw [inf_assoc]
ext X Y f
change _ ∧ _ ∧ _... | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | IsProper.eq_valuativeCriterion | The **valuative criterion** for proper morphisms. |
@[stacks 0BX5]
IsProper.of_valuativeCriterion [QuasiCompact f] [QuasiSeparated f] [LocallyOfFiniteType f]
(H : ValuativeCriterion f) : IsProper f := by
rw [eq_valuativeCriterion]
exact ⟨⟨⟨‹_›, ‹_›⟩, ‹_›⟩, ‹_›⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Morphisms.Immersion",
"Mathlib.AlgebraicGeometry.Morphisms.Proper",
"Mathlib.RingTheory.RingHom.Injective",
"Mathlib.RingTheory.Valuation.LocalSubring"
] | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | IsProper.of_valuativeCriterion | The **valuative criterion** for proper morphisms. |
@[simp]
objD (n : ℕ) : X _⦋n + 1⦌ ⟶ X _⦋n⦌ :=
∑ i : Fin (n + 2), (-1 : ℤ) ^ (i : ℕ) • X.δ i
/-! | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | objD | The differential on the alternating face map complex is the alternate
sum of the face maps |
d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by
dsimp
simp only [comp_sum, sum_comp, ← Finset.sum_product']
let P := Fin (n + 2) × Fin (n + 3)
let S : Finset P := {ij : P | (ij.2 : ℕ) ≤ (ij.1 : ℕ)}
rw [Finset.univ_product_univ, ← Finset.sum_add_sum_compl S, ← eq_neg_iff_add_eq_zero,
← Finset.sum_n... | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | d_squared | null |
obj : ChainComplex C ℕ :=
ChainComplex.of (fun n => X _⦋n⦌) (objD X) (d_squared X)
@[simp] | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | obj | The alternating face map complex, on objects |
obj_X (X : SimplicialObject C) (n : ℕ) : (AlternatingFaceMapComplex.obj X).X n = X _⦋n⦌ :=
rfl
@[simp] | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | obj_X | null |
obj_d_eq (X : SimplicialObject C) (n : ℕ) :
(AlternatingFaceMapComplex.obj X).d (n + 1) n
= ∑ i : Fin (n + 2), (-1 : ℤ) ^ (i : ℕ) • X.δ i := by
apply ChainComplex.of_d
variable {X} {Y} | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | obj_d_eq | null |
map (f : X ⟶ Y) : obj X ⟶ obj Y :=
ChainComplex.ofHom _ _ _ _ _ _ (fun n => f.app (op ⦋n⦌)) fun n => by
dsimp
rw [comp_sum, sum_comp]
refine Finset.sum_congr rfl fun _ _ => ?_
rw [comp_zsmul, zsmul_comp]
congr 1
symm
apply f.naturality
@[simp] | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | map | The alternating face map complex, on morphisms |
map_f (f : X ⟶ Y) (n : ℕ) : (map f).f n = f.app (op ⦋n⦌) :=
rfl | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | map_f | null |
alternatingFaceMapComplex : SimplicialObject C ⥤ ChainComplex C ℕ where
obj := AlternatingFaceMapComplex.obj
map f := AlternatingFaceMapComplex.map f
variable {C}
@[simp] | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | alternatingFaceMapComplex | The alternating face map complex, as a functor |
alternatingFaceMapComplex_obj_X (X : SimplicialObject C) (n : ℕ) :
((alternatingFaceMapComplex C).obj X).X n = X _⦋n⦌ :=
rfl
@[simp] | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | alternatingFaceMapComplex_obj_X | null |
alternatingFaceMapComplex_obj_d (X : SimplicialObject C) (n : ℕ) :
((alternatingFaceMapComplex C).obj X).d (n + 1) n = AlternatingFaceMapComplex.objD X n := by
dsimp only [alternatingFaceMapComplex, AlternatingFaceMapComplex.obj]
apply ChainComplex.of_d
@[simp] | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | alternatingFaceMapComplex_obj_d | null |
alternatingFaceMapComplex_map_f {X Y : SimplicialObject C} (f : X ⟶ Y) (n : ℕ) :
((alternatingFaceMapComplex C).map f).f n = f.app (op ⦋n⦌) :=
rfl | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | alternatingFaceMapComplex_map_f | null |
map_alternatingFaceMapComplex {D : Type*} [Category D] [Preadditive D] (F : C ⥤ D)
[F.Additive] :
alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex _ =
(SimplicialObject.whiskering C D).obj F ⋙ alternatingFaceMapComplex D := by
apply CategoryTheory.Functor.ext
· intro X Y f
ext n
simp only... | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | map_alternatingFaceMapComplex | null |
karoubi_alternatingFaceMapComplex_d (P : Karoubi (SimplicialObject C)) (n : ℕ) :
((AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)).d (n + 1) n).f =
P.p.app (op ⦋n + 1⦌) ≫ (AlternatingFaceMapComplex.obj P.X).d (n + 1) n := by
dsimp
simp only [AlternatingFaceMapComplex.obj_d_eq, Karou... | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | karoubi_alternatingFaceMapComplex_d | null |
ε [Limits.HasZeroObject C] :
SimplicialObject.Augmented.drop ⋙ AlgebraicTopology.alternatingFaceMapComplex C ⟶
SimplicialObject.Augmented.point ⋙ ChainComplex.single₀ C where
app X := by
refine (ChainComplex.toSingle₀Equiv _ _).symm ?_
refine ⟨X.hom.app (op ⦋0⦌), ?_⟩
dsimp
rw [alternatingFac... | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | ε | The natural transformation which gives the augmentation of the alternating face map
complex attached to an augmented simplicial object. |
ε_app_f_zero [Limits.HasZeroObject C] (X : SimplicialObject.Augmented C) :
(ε.app X).f 0 = X.hom.app (op ⦋0⦌) :=
ChainComplex.toSingle₀Equiv_symm_apply_f_zero _ _
@[simp] | lemma | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | ε_app_f_zero | null |
ε_app_f_succ [Limits.HasZeroObject C] (X : SimplicialObject.Augmented C) (n : ℕ) :
(ε.app X).f (n + 1) = 0 := rfl | lemma | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | ε_app_f_succ | null |
inclusionOfMooreComplexMap (X : SimplicialObject A) :
(normalizedMooreComplex A).obj X ⟶ (alternatingFaceMapComplex A).obj X := by
dsimp only [normalizedMooreComplex, NormalizedMooreComplex.obj,
alternatingFaceMapComplex, AlternatingFaceMapComplex.obj]
apply ChainComplex.ofHom _ _ _ _ _ _ (fun n => (Normali... | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | inclusionOfMooreComplexMap | The inclusion map of the Moore complex in the alternating face map complex |
inclusionOfMooreComplexMap_f (X : SimplicialObject A) (n : ℕ) :
(inclusionOfMooreComplexMap X).f n = (NormalizedMooreComplex.objX X n).arrow := by
dsimp only [inclusionOfMooreComplexMap]
exact ChainComplex.ofHom_f _ _ _ _ _ _ _ _ n
variable (A) | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | inclusionOfMooreComplexMap_f | null |
@[simps]
inclusionOfMooreComplex : normalizedMooreComplex A ⟶ alternatingFaceMapComplex A where
app := inclusionOfMooreComplexMap | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | inclusionOfMooreComplex | The inclusion map of the Moore complex in the alternating face map complex,
as a natural transformation |
@[simp]
objD (n : ℕ) : X.obj ⦋n⦌ ⟶ X.obj ⦋n + 1⦌ :=
∑ i : Fin (n + 2), (-1 : ℤ) ^ (i : ℕ) • X.δ i | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | objD | The differential on the alternating coface map complex is the alternate
sum of the coface maps |
d_eq_unop_d (n : ℕ) :
objD X n =
(AlternatingFaceMapComplex.objD ((cosimplicialSimplicialEquiv C).functor.obj (op X))
n).unop := by
simp only [objD, AlternatingFaceMapComplex.objD, unop_sum, unop_zsmul]
rfl | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | d_eq_unop_d | null |
d_squared (n : ℕ) : objD X n ≫ objD X (n + 1) = 0 := by
simp only [d_eq_unop_d, ← unop_comp, AlternatingFaceMapComplex.d_squared, unop_zero] | theorem | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | d_squared | null |
obj : CochainComplex C ℕ :=
CochainComplex.of (fun n => X.obj ⦋n⦌) (objD X) (d_squared X)
variable {X} {Y} | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | obj | The alternating coface map complex, on objects |
@[simp]
map (f : X ⟶ Y) : obj X ⟶ obj Y :=
CochainComplex.ofHom _ _ _ _ _ _ (fun n => f.app ⦋n⦌) fun n => by
dsimp
rw [comp_sum, sum_comp]
refine Finset.sum_congr rfl fun x _ => ?_
rw [comp_zsmul, zsmul_comp]
congr 1
symm
apply f.naturality | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | map | The alternating face map complex, on morphisms |
@[simps]
alternatingCofaceMapComplex : CosimplicialObject C ⥤ CochainComplex C ℕ where
obj := AlternatingCofaceMapComplex.obj
map f := AlternatingCofaceMapComplex.map f | def | AlgebraicTopology | [
"Mathlib.Algebra.Homology.Additive",
"Mathlib.AlgebraicTopology.MooreComplex",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.CategoryTheory.Preadditive.Opposite",
"Mathlib.CategoryTheory.Idempotents.FunctorCategories"
] | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | alternatingCofaceMapComplex | The alternating coface map complex, as a functor |
@[simps]
cechNerve : SimplicialObject C where
obj n := widePullback.{0} f.right (fun _ : Fin (n.unop.len + 1) => f.left) fun _ => f.hom
map g := WidePullback.lift (WidePullback.base _)
(fun i => WidePullback.π _ (g.unop.toOrderHom i)) (by simp) | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | cechNerve | The Čech nerve associated to an arrow. |
@[simps]
mapCechNerve {f g : Arrow C}
[∀ n : ℕ, HasWidePullback f.right (fun _ : Fin (n + 1) => f.left) fun _ => f.hom]
[∀ n : ℕ, HasWidePullback g.right (fun _ : Fin (n + 1) => g.left) fun _ => g.hom] (F : f ⟶ g) :
f.cechNerve ⟶ g.cechNerve where
app n :=
WidePullback.lift (WidePullback.base _ ≫ F.ri... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | mapCechNerve | The morphism between Čech nerves associated to a morphism of arrows. |
@[simps]
augmentedCechNerve : SimplicialObject.Augmented C where
left := f.cechNerve
right := f.right
hom := { app := fun _ => WidePullback.base _ } | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | augmentedCechNerve | The augmented Čech nerve associated to an arrow. |
@[simps]
mapAugmentedCechNerve {f g : Arrow C}
[∀ n : ℕ, HasWidePullback f.right (fun _ : Fin (n + 1) => f.left) fun _ => f.hom]
[∀ n : ℕ, HasWidePullback g.right (fun _ : Fin (n + 1) => g.left) fun _ => g.hom] (F : f ⟶ g) :
f.augmentedCechNerve ⟶ g.augmentedCechNerve where
left := mapCechNerve F
right ... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | mapAugmentedCechNerve | The morphism between augmented Čech nerve associated to a morphism of arrows. |
@[simps]
cechNerve : Arrow C ⥤ SimplicialObject C where
obj f := f.cechNerve
map F := Arrow.mapCechNerve F | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | cechNerve | The Čech nerve construction, as a functor from `Arrow C`. |
@[simps!]
augmentedCechNerve : Arrow C ⥤ SimplicialObject.Augmented C where
obj f := f.augmentedCechNerve
map F := Arrow.mapAugmentedCechNerve F | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | augmentedCechNerve | The augmented Čech nerve construction, as a functor from `Arrow C`. |
@[simps]
equivalenceRightToLeft (X : SimplicialObject.Augmented C) (F : Arrow C)
(G : X ⟶ F.augmentedCechNerve) : Augmented.toArrow.obj X ⟶ F where
left := G.left.app _ ≫ WidePullback.π _ 0
right := G.right
w := by
have := G.w
apply_fun fun e => e.app (Opposite.op ⦋0⦌) at this
simpa using this | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | equivalenceRightToLeft | A helper function used in defining the Čech adjunction. |
@[simps]
equivalenceLeftToRight (X : SimplicialObject.Augmented C) (F : Arrow C)
(G : Augmented.toArrow.obj X ⟶ F) : X ⟶ F.augmentedCechNerve where
left :=
{ app := fun x =>
Limits.WidePullback.lift (X.hom.app _ ≫ G.right)
(fun i => X.left.map (SimplexCategory.const _ x.unop i).op ≫ G.left) ... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | equivalenceLeftToRight | A helper function used in defining the Čech adjunction. |
@[simps]
cechNerveEquiv (X : SimplicialObject.Augmented C) (F : Arrow C) :
(Augmented.toArrow.obj X ⟶ F) ≃ (X ⟶ F.augmentedCechNerve) where
toFun := equivalenceLeftToRight _ _
invFun := equivalenceRightToLeft _ _
left_inv A := by ext <;> simp
right_inv := by
intro A
ext x : 2
· refine WidePullba... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | cechNerveEquiv | A helper function used in defining the Čech adjunction. |
cechNerveAdjunction : (Augmented.toArrow : _ ⥤ Arrow C) ⊣ augmentedCechNerve :=
Adjunction.mkOfHomEquiv
{ homEquiv := cechNerveEquiv
homEquiv_naturality_left_symm := by dsimp [cechNerveEquiv]; cat_disch
homEquiv_naturality_right := by
dsimp [cechNerveEquiv]
intro X Y Y' f g
cha... | abbrev | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | cechNerveAdjunction | The augmented Čech nerve construction is right adjoint to the `toArrow` functor. |
@[simps]
cechConerve : CosimplicialObject C where
obj n := widePushout f.left (fun _ : Fin (n.len + 1) => f.right) fun _ => f.hom
map {x y} g := by
refine WidePushout.desc (WidePushout.head _)
(fun i => (@WidePushout.ι _ _ _ _ _ (fun _ => f.hom) (_) (g.toOrderHom i))) (fun j => ?_)
rw [← WidePushout.a... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | cechConerve | The Čech conerve associated to an arrow. |
@[simps]
mapCechConerve {f g : Arrow C}
[∀ n : ℕ, HasWidePushout f.left (fun _ : Fin (n + 1) => f.right) fun _ => f.hom]
[∀ n : ℕ, HasWidePushout g.left (fun _ : Fin (n + 1) => g.right) fun _ => g.hom] (F : f ⟶ g) :
f.cechConerve ⟶ g.cechConerve where
app n := WidePushout.desc (F.left ≫ WidePushout.head _... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | mapCechConerve | The morphism between Čech conerves associated to a morphism of arrows. |
@[simps]
augmentedCechConerve : CosimplicialObject.Augmented C where
left := f.left
right := f.cechConerve
hom :=
{ app := fun _ => (WidePushout.head _ : f.left ⟶ _) } | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | augmentedCechConerve | The augmented Čech conerve associated to an arrow. |
@[simps]
mapAugmentedCechConerve {f g : Arrow C}
[∀ n : ℕ, HasWidePushout f.left (fun _ : Fin (n + 1) => f.right) fun _ => f.hom]
[∀ n : ℕ, HasWidePushout g.left (fun _ : Fin (n + 1) => g.right) fun _ => g.hom] (F : f ⟶ g) :
f.augmentedCechConerve ⟶ g.augmentedCechConerve where
left := F.left
right := m... | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | mapAugmentedCechConerve | The morphism between augmented Čech conerves associated to a morphism of arrows. |
@[simps]
cechConerve : Arrow C ⥤ CosimplicialObject C where
obj f := f.cechConerve
map F := Arrow.mapCechConerve F | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | cechConerve | The Čech conerve construction, as a functor from `Arrow C`. |
@[simps]
augmentedCechConerve : Arrow C ⥤ CosimplicialObject.Augmented C where
obj f := f.augmentedCechConerve
map F := Arrow.mapAugmentedCechConerve F | def | AlgebraicTopology | [
"Mathlib.AlgebraicTopology.SimplicialObject.Basic",
"Mathlib.CategoryTheory.Comma.Arrow",
"Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks",
"Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts"
] | Mathlib/AlgebraicTopology/CechNerve.lean | augmentedCechConerve | The augmented Čech conerve construction, as a functor from `Arrow C`. |
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