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 ⌀ |
|---|---|---|---|---|---|---|
@[simp]
default_asIdeal {K} [Field K] : (default : Spec(K)).asIdeal = ⊥ := rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | default_asIdeal | null |
basicOpen : X.Opens :=
X.toLocallyRingedSpace.toRingedSpace.basicOpen f | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen | The subset of the underlying space where the given section does not vanish. |
mem_basicOpen (x : X) (hx : x ∈ U) :
x ∈ X.basicOpen f ↔ IsUnit (X.presheaf.germ U x hx f) :=
RingedSpace.mem_basicOpen _ _ _ _ | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | mem_basicOpen | null |
@[simp]
mem_basicOpen' (x : U) : ↑x ∈ X.basicOpen f ↔ IsUnit (X.presheaf.germ U x x.2 f) :=
RingedSpace.mem_basicOpen _ _ _ _ | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | mem_basicOpen' | A variant of `mem_basicOpen` for bundled `x : U`. |
mem_basicOpen'' {U : X.Opens} (f : Γ(X, U)) (x : X) :
x ∈ X.basicOpen f ↔ ∃ (m : x ∈ U), IsUnit (X.presheaf.germ U x m f) :=
Iff.rfl
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | mem_basicOpen'' | A variant of `mem_basicOpen` without the `x ∈ U` assumption. |
mem_basicOpen_top (f : Γ(X, ⊤)) (x : X) :
x ∈ X.basicOpen f ↔ IsUnit (X.presheaf.germ ⊤ x trivial f) :=
RingedSpace.mem_top_basicOpen _ f x
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | mem_basicOpen_top | null |
basicOpen_res (i : op U ⟶ op V) : X.basicOpen (X.presheaf.map i f) = V ⊓ X.basicOpen f :=
RingedSpace.basicOpen_res _ i f
@[simp 1100] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_res | null |
basicOpen_res_eq (i : op U ⟶ op V) [IsIso i] :
X.basicOpen (X.presheaf.map i f) = X.basicOpen f :=
RingedSpace.basicOpen_res_eq _ i f
@[sheaf_restrict] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_res_eq | null |
basicOpen_le : X.basicOpen f ≤ U :=
RingedSpace.basicOpen_le _ _
@[sheaf_restrict] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_le | null |
basicOpen_restrict (i : V ⟶ U) (f : Γ(X, U)) :
X.basicOpen (TopCat.Presheaf.restrict f i) ≤ X.basicOpen f :=
(Scheme.basicOpen_res _ _ _).trans_le inf_le_right
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_restrict | null |
preimage_basicOpen {X Y : Scheme.{u}} (f : X ⟶ Y) {U : Y.Opens} (r : Γ(Y, U)) :
f ⁻¹ᵁ (Y.basicOpen r) = X.basicOpen (f.app U r) :=
LocallyRingedSpace.preimage_basicOpen f.toLRSHom r | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | preimage_basicOpen | null |
preimage_basicOpen_top {X Y : Scheme.{u}} (f : X ⟶ Y) (r : Γ(Y, ⊤)) :
f ⁻¹ᵁ (Y.basicOpen r) = X.basicOpen (f.appTop r) :=
preimage_basicOpen .. | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | preimage_basicOpen_top | null |
basicOpen_appLE {X Y : Scheme.{u}} (f : X ⟶ Y) (U : X.Opens) (V : Y.Opens) (e : U ≤ f ⁻¹ᵁ V)
(s : Γ(Y, V)) : X.basicOpen (f.appLE V U e s) = U ⊓ f ⁻¹ᵁ (Y.basicOpen s) := by
simp only [preimage_basicOpen, Hom.appLE, CommRingCat.comp_apply]
rw [basicOpen_res]
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_appLE | null |
basicOpen_zero (U : X.Opens) : X.basicOpen (0 : Γ(X, U)) = ⊥ :=
LocallyRingedSpace.basicOpen_zero _ U
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_zero | null |
basicOpen_mul : X.basicOpen (f * g) = X.basicOpen f ⊓ X.basicOpen g :=
RingedSpace.basicOpen_mul _ _ _ | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_mul | null |
basicOpen_pow {n : ℕ} (h : 0 < n) : X.basicOpen (f ^ n) = X.basicOpen f :=
RingedSpace.basicOpen_pow _ _ _ h | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_pow | null |
basicOpen_add_le :
X.basicOpen (f + g) ≤ X.basicOpen f ⊔ X.basicOpen g := by
intro x hx
have hxU : x ∈ U := X.basicOpen_le _ hx
simp only [SetLike.mem_coe, Scheme.mem_basicOpen _ _ _ hxU, map_add, Opens.coe_sup,
Set.mem_union] at hx ⊢
exact IsLocalRing.isUnit_or_isUnit_of_isUnit_add hx | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_add_le | null |
basicOpen_of_isUnit {f : Γ(X, U)} (hf : IsUnit f) : X.basicOpen f = U :=
RingedSpace.basicOpen_of_isUnit _ hf
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_of_isUnit | null |
basicOpen_one : X.basicOpen (1 : Γ(X, U)) = U :=
X.basicOpen_of_isUnit isUnit_one | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_one | null |
algebra_section_section_basicOpen {X : Scheme} {U : X.Opens} (f : Γ(X, U)) :
Algebra Γ(X, U) Γ(X, X.basicOpen f) :=
(X.presheaf.map (homOfLE <| X.basicOpen_le f : _ ⟶ U).op).hom.toAlgebra | instance | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | algebra_section_section_basicOpen | null |
zeroLocus {U : X.Opens} (s : Set Γ(X, U)) : Set X := X.toRingedSpace.zeroLocus s | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus | The zero locus of a set of sections `s` over an open set `U` is the closed set consisting of
the complement of `U` and of all points of `U`, where all elements of `f` vanish. |
zeroLocus_def {U : X.Opens} (s : Set Γ(X, U)) :
X.zeroLocus s = ⋂ f ∈ s, (X.basicOpen f).carrierᶜ :=
rfl | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_def | null |
zeroLocus_isClosed {U : X.Opens} (s : Set Γ(X, U)) :
IsClosed (X.zeroLocus s) :=
X.toRingedSpace.zeroLocus_isClosed s | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_isClosed | null |
zeroLocus_singleton {U : X.Opens} (f : Γ(X, U)) :
X.zeroLocus {f} = (↑(X.basicOpen f))ᶜ :=
X.toRingedSpace.zeroLocus_singleton f
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_singleton | null |
zeroLocus_empty_eq_univ {U : X.Opens} :
X.zeroLocus (∅ : Set Γ(X, U)) = Set.univ :=
X.toRingedSpace.zeroLocus_empty_eq_univ
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_empty_eq_univ | null |
mem_zeroLocus_iff {U : X.Opens} (s : Set Γ(X, U)) (x : X) :
x ∈ X.zeroLocus s ↔ ∀ f ∈ s, x ∉ X.basicOpen f :=
X.toRingedSpace.mem_zeroLocus_iff s x | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | mem_zeroLocus_iff | null |
codisjoint_zeroLocus {U : X.Opens}
(s : Set Γ(X, U)) : Codisjoint (X.zeroLocus s) U := by
have (x : X) : ∀ f ∈ s, x ∈ X.basicOpen f → x ∈ U := fun _ _ h ↦ X.basicOpen_le _ h
simpa [codisjoint_iff_le_sup, Set.ext_iff, or_iff_not_imp_left] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | codisjoint_zeroLocus | null |
zeroLocus_span {U : X.Opens} (s : Set Γ(X, U)) :
X.zeroLocus (U := U) (Ideal.span s) = X.zeroLocus s := by
ext x
simp only [Scheme.mem_zeroLocus_iff, SetLike.mem_coe]
refine ⟨fun H f hfs ↦ H f (Ideal.subset_span hfs), fun H f ↦ Submodule.span_induction H ?_ ?_ ?_⟩
· simp only [Scheme.basicOpen_zero]; exact not_false
· exact fun a b _ _ ha hb H ↦ (X.basicOpen_add_le a b H).elim ha hb
· simp +contextual | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_span | null |
zeroLocus_map {U V : X.Opens} (i : U ≤ V) (s : Set Γ(X, V)) :
X.zeroLocus ((X.presheaf.map (homOfLE i).op).hom '' s) = X.zeroLocus s ∪ Uᶜ := by
ext x
suffices (∀ f ∈ s, x ∈ U → x ∉ X.basicOpen f) ↔ x ∈ U → (∀ f ∈ s, x ∉ X.basicOpen f) by
simpa [or_iff_not_imp_right]
grind | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_map | null |
zeroLocus_map_of_eq {U V : X.Opens} (i : U = V) (s : Set Γ(X, V)) :
X.zeroLocus ((X.presheaf.map (eqToHom i).op).hom '' s) = X.zeroLocus s := by
ext; simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_map_of_eq | null |
zeroLocus_mono {U : X.Opens} {s t : Set Γ(X, U)} (h : s ⊆ t) :
X.zeroLocus t ⊆ X.zeroLocus s := by
simp only [Set.subset_def, Scheme.mem_zeroLocus_iff]
exact fun x H f hf hxf ↦ H f (h hf) hxf | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_mono | null |
preimage_zeroLocus {X Y : Scheme.{u}} (f : X ⟶ Y) {U : Y.Opens} (s : Set Γ(Y, U)) :
f.base ⁻¹' Y.zeroLocus s = X.zeroLocus ((f.app U).hom '' s) := by
ext
simp [← Scheme.preimage_basicOpen]
rfl
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | preimage_zeroLocus | null |
zeroLocus_univ {U : X.Opens} :
X.zeroLocus (U := U) Set.univ = (↑U)ᶜ := by
ext x
simp only [Scheme.mem_zeroLocus_iff, Set.mem_univ, forall_const, Set.mem_compl_iff,
SetLike.mem_coe, ← not_exists, not_iff_not]
exact ⟨fun ⟨f, hf⟩ ↦ X.basicOpen_le f hf, fun _ ↦ ⟨1, by rwa [X.basicOpen_of_isUnit isUnit_one]⟩⟩ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_univ | null |
zeroLocus_iUnion {U : X.Opens} {ι : Type*} (f : ι → Set Γ(X, U)) :
X.zeroLocus (⋃ i, f i) = ⋂ i, X.zeroLocus (f i) := by
simpa [zeroLocus, AlgebraicGeometry.RingedSpace.zeroLocus] using Set.iInter_comm _ | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_iUnion | null |
zeroLocus_radical {U : X.Opens} (I : Ideal Γ(X, U)) :
X.zeroLocus (U := U) I.radical = X.zeroLocus (U := U) I := by
refine (X.zeroLocus_mono I.le_radical).antisymm ?_
simp only [Set.subset_def, mem_zeroLocus_iff, SetLike.mem_coe]
rintro x H f ⟨n, hn⟩ hx
rcases n.eq_zero_or_pos with rfl | hn'
· exact H f (by simpa using I.mul_mem_left f hn) hx
· exact H _ hn (X.basicOpen_pow f hn' ▸ hx) | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | zeroLocus_radical | null |
basicOpen_eq_of_affine {R : CommRingCat} (f : R) :
(Spec R).basicOpen ((Scheme.ΓSpecIso R).inv f) = PrimeSpectrum.basicOpen f := by
ext x
simp only [SetLike.mem_coe, Scheme.mem_basicOpen_top]
suffices IsUnit (StructureSheaf.toStalk R x f) ↔ f ∉ PrimeSpectrum.asIdeal x by exact this
rw [← isUnit_map_iff (StructureSheaf.stalkToFiberRingHom R x).hom,
StructureSheaf.stalkToFiberRingHom_toStalk]
exact
(IsLocalization.AtPrime.isUnit_to_map_iff (Localization.AtPrime (PrimeSpectrum.asIdeal x))
(PrimeSpectrum.asIdeal x) f :
_)
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_eq_of_affine | null |
basicOpen_eq_of_affine' {R : CommRingCat} (f : Γ(Spec R, ⊤)) :
(Spec R).basicOpen f = PrimeSpectrum.basicOpen ((Scheme.ΓSpecIso R).hom f) := by
convert basicOpen_eq_of_affine ((Scheme.ΓSpecIso R).hom f)
exact (Iso.hom_inv_id_apply (Scheme.ΓSpecIso R) f).symm | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | basicOpen_eq_of_affine' | null |
Scheme.Spec_map_presheaf_map_eqToHom {X : Scheme} {U V : X.Opens} (h : U = V) (W) :
(Spec.map (X.presheaf.map (eqToHom h).op)).app W = eqToHom (by cases h; simp) := by
have : Scheme.Spec.map (X.presheaf.map (𝟙 (op U))).op = 𝟙 _ := by
rw [X.presheaf.map_id, op_id, Scheme.Spec.map_id]
cases h
refine (Scheme.congr_app this _).trans ?_
simp [eqToHom_map] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | Scheme.Spec_map_presheaf_map_eqToHom | null |
germ_eq_zero_of_pow_mul_eq_zero {X : Scheme.{u}} {U : Opens X} (x : U) {f s : Γ(X, U)}
(hx : x.val ∈ X.basicOpen s) {n : ℕ} (hf : s ^ n * f = 0) : X.presheaf.germ U x x.2 f = 0 := by
rw [Scheme.mem_basicOpen X s x x.2] at hx
have hu : IsUnit (X.presheaf.germ _ x x.2 (s ^ n)) := by
rw [map_pow]
exact IsUnit.pow n hx
rw [← hu.mul_right_eq_zero, ← map_mul, hf, map_zero]
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | germ_eq_zero_of_pow_mul_eq_zero | null |
Scheme.iso_hom_base_inv_base {X Y : Scheme.{u}} (e : X ≅ Y) :
e.hom.base ≫ e.inv.base = 𝟙 _ :=
LocallyRingedSpace.iso_hom_base_inv_base (Scheme.forgetToLocallyRingedSpace.mapIso e)
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | Scheme.iso_hom_base_inv_base | null |
Scheme.iso_hom_base_inv_base_apply {X Y : Scheme.{u}} (e : X ≅ Y) (x : X) :
(e.inv.base (e.hom.base x)) = x := by
change (e.hom.base ≫ e.inv.base) x = 𝟙 X.toPresheafedSpace x
simp
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | Scheme.iso_hom_base_inv_base_apply | null |
Scheme.iso_inv_base_hom_base {X Y : Scheme.{u}} (e : X ≅ Y) :
e.inv.base ≫ e.hom.base = 𝟙 _ :=
LocallyRingedSpace.iso_inv_base_hom_base (Scheme.forgetToLocallyRingedSpace.mapIso e)
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | Scheme.iso_inv_base_hom_base | null |
Scheme.iso_inv_base_hom_base_apply {X Y : Scheme.{u}} (e : X ≅ Y) (y : Y) :
(e.hom.base (e.inv.base y)) = y := by
change (e.inv.base ≫ e.hom.base) y = 𝟙 Y.toPresheafedSpace y
simp | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | Scheme.iso_inv_base_hom_base_apply | null |
Spec_zeroLocus_eq_zeroLocus {R : CommRingCat} (s : Set R) :
(Spec R).zeroLocus ((Scheme.ΓSpecIso R).inv '' s) = PrimeSpectrum.zeroLocus s := by
ext x
suffices (∀ a ∈ s, x ∉ PrimeSpectrum.basicOpen a) ↔ x ∈ PrimeSpectrum.zeroLocus s by simpa
simp [Spec_carrier, PrimeSpectrum.mem_zeroLocus, Set.subset_def,
PrimeSpectrum.mem_basicOpen _ x]
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | Spec_zeroLocus_eq_zeroLocus | null |
Spec_zeroLocus {R : CommRingCat} (s : Set Γ(Spec R, ⊤)) :
(Spec R).zeroLocus s = PrimeSpectrum.zeroLocus ((Scheme.ΓSpecIso R).inv ⁻¹' s) := by
convert Spec_zeroLocus_eq_zeroLocus ((Scheme.ΓSpecIso R).inv ⁻¹' s)
rw [Set.image_preimage_eq]
exact (ConcreteCategory.bijective_of_isIso (C := CommRingCat) _).2 | theorem | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | Spec_zeroLocus | null |
@[simp]
stalkMap_id (X : Scheme.{u}) (x : X) :
(𝟙 X : X ⟶ X).stalkMap x = 𝟙 (X.presheaf.stalk x) :=
PresheafedSpace.stalkMap.id _ x | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_id | null |
stalkMap_comp {X Y Z : Scheme.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g : X ⟶ Z).stalkMap x = g.stalkMap (f.base x) ≫ f.stalkMap x :=
PresheafedSpace.stalkMap.comp f.toPshHom g.toPshHom x
@[reassoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_comp | null |
stalkSpecializes_stalkMap (x x' : X)
(h : x ⤳ x') : Y.presheaf.stalkSpecializes (f.base.hom.map_specializes h) ≫ f.stalkMap x =
f.stalkMap x' ≫ X.presheaf.stalkSpecializes h :=
PresheafedSpace.stalkMap.stalkSpecializes_stalkMap f.toPshHom h | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkSpecializes_stalkMap | null |
stalkSpecializes_stalkMap_apply (x x' : X) (h : x ⤳ x') (y) :
f.stalkMap x (Y.presheaf.stalkSpecializes (f.base.hom.map_specializes h) y) =
(X.presheaf.stalkSpecializes h (f.stalkMap x' y)) :=
DFunLike.congr_fun (CommRingCat.hom_ext_iff.mp (stalkSpecializes_stalkMap f x x' h)) y
@[reassoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkSpecializes_stalkMap_apply | null |
stalkMap_congr (f g : X ⟶ Y) (hfg : f = g) (x x' : X)
(hxx' : x = x') : f.stalkMap x ≫ (X.presheaf.stalkCongr (.of_eq hxx')).hom =
(Y.presheaf.stalkCongr (.of_eq <| hfg ▸ hxx' ▸ rfl)).hom ≫ g.stalkMap x' :=
LocallyRingedSpace.stalkMap_congr f.toLRSHom g.toLRSHom congr(($hfg).toLRSHom) x x' hxx'
@[reassoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_congr | null |
stalkMap_congr_hom (f g : X ⟶ Y) (hfg : f = g) (x : X) :
f.stalkMap x = (Y.presheaf.stalkCongr (.of_eq <| hfg ▸ rfl)).hom ≫ g.stalkMap x :=
LocallyRingedSpace.stalkMap_congr_hom f.toLRSHom g.toLRSHom congr(($hfg).toLRSHom) x
@[reassoc] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_congr_hom | null |
stalkMap_congr_point (x x' : X) (hxx' : x = x') :
f.stalkMap x ≫ (X.presheaf.stalkCongr (.of_eq hxx')).hom =
(Y.presheaf.stalkCongr (.of_eq <| hxx' ▸ rfl)).hom ≫ f.stalkMap x' :=
LocallyRingedSpace.stalkMap_congr_point f.toLRSHom x x' hxx'
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_congr_point | null |
stalkMap_hom_inv (e : X ≅ Y) (y : Y) :
e.hom.stalkMap (e.inv.base y) ≫ e.inv.stalkMap y =
(Y.presheaf.stalkCongr (.of_eq (by simp))).hom :=
LocallyRingedSpace.stalkMap_hom_inv (forgetToLocallyRingedSpace.mapIso e) y
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_hom_inv | null |
stalkMap_hom_inv_apply (e : X ≅ Y) (y : Y) (z) :
e.inv.stalkMap y (e.hom.stalkMap (e.inv.base y) z) =
(Y.presheaf.stalkCongr (.of_eq (by simp))).hom z :=
DFunLike.congr_fun (CommRingCat.hom_ext_iff.mp (stalkMap_hom_inv e y)) z
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_hom_inv_apply | null |
stalkMap_inv_hom (e : X ≅ Y) (x : X) :
e.inv.stalkMap (e.hom.base x) ≫ e.hom.stalkMap x =
(X.presheaf.stalkCongr (.of_eq (by simp))).hom :=
LocallyRingedSpace.stalkMap_inv_hom (forgetToLocallyRingedSpace.mapIso e) x
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_inv_hom | null |
stalkMap_inv_hom_apply (e : X ≅ Y) (x : X) (y) :
e.hom.stalkMap x (e.inv.stalkMap (e.hom.base x) y) =
(X.presheaf.stalkCongr (.of_eq (by simp))).hom y :=
DFunLike.congr_fun (CommRingCat.hom_ext_iff.mp (stalkMap_inv_hom e x)) y
@[reassoc (attr := simp)] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_inv_hom_apply | null |
stalkMap_germ (U : Y.Opens) (x : X) (hx : f.base x ∈ U) :
Y.presheaf.germ U (f.base x) hx ≫ f.stalkMap x =
f.app U ≫ X.presheaf.germ (f ⁻¹ᵁ U) x hx :=
PresheafedSpace.stalkMap_germ f.toPshHom U x hx
@[simp] | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_germ | null |
stalkMap_germ_apply (U : Y.Opens) (x : X) (hx : f.base x ∈ U) (y) :
f.stalkMap x (Y.presheaf.germ _ (f.base x) hx y) =
X.presheaf.germ (f ⁻¹ᵁ U) x hx (f.app U y) :=
PresheafedSpace.stalkMap_germ_apply f.toPshHom U x hx y | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | stalkMap_germ_apply | null |
noncomputable arrowStalkMapIsoOfEq {x y : X}
(h : x = y) : Arrow.mk (f.stalkMap x) ≅ Arrow.mk (f.stalkMap y) :=
Arrow.isoMk (Y.presheaf.stalkCongr <| (Inseparable.of_eq h).map f.continuous)
(X.presheaf.stalkCongr <| Inseparable.of_eq h) <| by
simp only [Arrow.mk_left, Arrow.mk_right, Functor.id_obj, TopCat.Presheaf.stalkCongr_hom,
Arrow.mk_hom]
rw [Scheme.stalkSpecializes_stalkMap] | def | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | arrowStalkMapIsoOfEq | If `x = y`, the stalk maps are isomorphic. |
@[simp]
Spec_closedPoint {R S : CommRingCat} [IsLocalRing R] [IsLocalRing S]
{f : R ⟶ S} [IsLocalHom f.hom] : (Spec.map f).base (closedPoint S) = closedPoint R :=
IsLocalRing.comap_closedPoint f.hom | lemma | AlgebraicGeometry | [
"Mathlib.AlgebraicGeometry.Spec",
"Mathlib.Algebra.Category.Ring.Constructions",
"Mathlib.CategoryTheory.Elementwise"
] | Mathlib/AlgebraicGeometry/Scheme.lean | Spec_closedPoint | null |
Spec.topObj (R : CommRingCat.{u}) : TopCat :=
TopCat.of (PrimeSpectrum R)
@[simp] theorem Spec.topObj_forget {R} : ToType (Spec.topObj R) = PrimeSpectrum R :=
rfl | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.topObj | The spectrum of a commutative ring, as a topological space. |
Spec.topMap {R S : CommRingCat.{u}} (f : R ⟶ S) : Spec.topObj S ⟶ Spec.topObj R :=
TopCat.ofHom (PrimeSpectrum.comap f.hom)
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.topMap | The induced map of a ring homomorphism on the ring spectra, as a morphism of topological spaces. |
Spec.topMap_id (R : CommRingCat.{u}) : Spec.topMap (𝟙 R) = 𝟙 (Spec.topObj R) :=
rfl
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.topMap_id | null |
Spec.topMap_comp {R S T : CommRingCat.{u}} (f : R ⟶ S) (g : S ⟶ T) :
Spec.topMap (f ≫ g) = Spec.topMap g ≫ Spec.topMap f :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.topMap_comp | null |
@[simps!]
Spec.toTop : CommRingCat.{u}ᵒᵖ ⥤ TopCat where
obj R := Spec.topObj (unop R)
map {_ _} f := Spec.topMap f.unop | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.toTop | The spectrum, as a contravariant functor from commutative rings to topological spaces. |
@[simps]
Spec.sheafedSpaceObj (R : CommRingCat.{u}) : SheafedSpace CommRingCat where
carrier := Spec.topObj R
presheaf := (structureSheaf R).1
IsSheaf := (structureSheaf R).2 | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.sheafedSpaceObj | The spectrum of a commutative ring, as a `SheafedSpace`. |
@[simps base c_app]
Spec.sheafedSpaceMap {R S : CommRingCat.{u}} (f : R ⟶ S) :
Spec.sheafedSpaceObj S ⟶ Spec.sheafedSpaceObj R where
base := Spec.topMap f
c :=
{ app := fun U => CommRingCat.ofHom <|
comap f.hom (unop U) ((TopologicalSpace.Opens.map (Spec.topMap f)).obj (unop U)) fun _ => id
naturality := fun {_ _} _ => by ext; rfl }
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.sheafedSpaceMap | The induced map of a ring homomorphism on the ring spectra, as a morphism of sheafed spaces. |
Spec.sheafedSpaceMap_id {R : CommRingCat.{u}} :
Spec.sheafedSpaceMap (𝟙 R) = 𝟙 (Spec.sheafedSpaceObj R) :=
AlgebraicGeometry.PresheafedSpace.Hom.ext _ _ (Spec.topMap_id R) <| by
ext
dsimp
rw [comap_id (by simp)]
simp
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.sheafedSpaceMap_id | null |
Spec.sheafedSpaceMap_comp {R S T : CommRingCat.{u}} (f : R ⟶ S) (g : S ⟶ T) :
Spec.sheafedSpaceMap (f ≫ g) = Spec.sheafedSpaceMap g ≫ Spec.sheafedSpaceMap f :=
AlgebraicGeometry.PresheafedSpace.Hom.ext _ _ (Spec.topMap_comp f g) <| by
ext
rw [NatTrans.comp_app, sheafedSpaceMap_c_app, Functor.whiskerRight_app, eqToHom_refl]
erw [(sheafedSpaceObj T).presheaf.map_id]
dsimp only [CommRingCat.hom_comp, RingHom.coe_comp, Function.comp_apply]
rw [comap_comp]
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.sheafedSpaceMap_comp | null |
@[simps]
Spec.toSheafedSpace : CommRingCat.{u}ᵒᵖ ⥤ SheafedSpace CommRingCat where
obj R := Spec.sheafedSpaceObj (unop R)
map f := Spec.sheafedSpaceMap f.unop
map_comp f g := by simp [Spec.sheafedSpaceMap_comp] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.toSheafedSpace | Spec, as a contravariant functor from commutative rings to sheafed spaces. |
Spec.toPresheafedSpace : CommRingCat.{u}ᵒᵖ ⥤ PresheafedSpace CommRingCat :=
Spec.toSheafedSpace ⋙ SheafedSpace.forgetToPresheafedSpace
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.toPresheafedSpace | Spec, as a contravariant functor from commutative rings to presheafed spaces. |
Spec.toPresheafedSpace_obj (R : CommRingCat.{u}ᵒᵖ) :
Spec.toPresheafedSpace.obj R = (Spec.sheafedSpaceObj (unop R)).toPresheafedSpace :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.toPresheafedSpace_obj | null |
Spec.toPresheafedSpace_obj_op (R : CommRingCat.{u}) :
Spec.toPresheafedSpace.obj (op R) = (Spec.sheafedSpaceObj R).toPresheafedSpace :=
rfl
@[simp] | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.toPresheafedSpace_obj_op | null |
Spec.toPresheafedSpace_map (R S : CommRingCat.{u}ᵒᵖ) (f : R ⟶ S) :
Spec.toPresheafedSpace.map f = Spec.sheafedSpaceMap f.unop :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.toPresheafedSpace_map | null |
Spec.toPresheafedSpace_map_op (R S : CommRingCat.{u}) (f : R ⟶ S) :
Spec.toPresheafedSpace.map f.op = Spec.sheafedSpaceMap f :=
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.toPresheafedSpace_map_op | null |
Spec.basicOpen_hom_ext {X : RingedSpace.{u}} {R : CommRingCat.{u}}
{α β : X ⟶ Spec.sheafedSpaceObj R} (w : α.base = β.base)
(h : ∀ r : R,
let U := PrimeSpectrum.basicOpen r
(toOpen R U ≫ α.c.app (op U)) ≫ X.presheaf.map (eqToHom (by rw [w])) =
toOpen R U ≫ β.c.app (op U)) :
α = β := by
ext : 1
· exact w
· apply
((TopCat.Sheaf.pushforward _ β.base).obj X.sheaf).hom_ext _ PrimeSpectrum.isBasis_basic_opens
intro r
apply (StructureSheaf.to_basicOpen_epi R r).1
simpa using h r | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.basicOpen_hom_ext | null |
@[simps! toSheafedSpace presheaf]
Spec.locallyRingedSpaceObj (R : CommRingCat.{u}) : LocallyRingedSpace :=
{ Spec.sheafedSpaceObj R with
isLocalRing := fun x =>
RingEquiv.isLocalRing (A := Localization.AtPrime x.asIdeal)
(Iso.commRingCatIsoToRingEquiv <| stalkIso R x).symm } | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.locallyRingedSpaceObj | The spectrum of a commutative ring, as a `LocallyRingedSpace`. |
Spec.locallyRingedSpaceObj_sheaf (R : CommRingCat.{u}) :
(Spec.locallyRingedSpaceObj R).sheaf = structureSheaf R := rfl | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.locallyRingedSpaceObj_sheaf | null |
Spec.locallyRingedSpaceObj_sheaf' (R : Type u) [CommRing R] :
(Spec.locallyRingedSpaceObj <| CommRingCat.of R).sheaf = structureSheaf R := rfl | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.locallyRingedSpaceObj_sheaf' | null |
Spec.locallyRingedSpaceObj_presheaf_map (R : CommRingCat.{u}) {U V} (i : U ⟶ V) :
(Spec.locallyRingedSpaceObj R).presheaf.map i =
(structureSheaf R).1.map i := rfl | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.locallyRingedSpaceObj_presheaf_map | null |
Spec.locallyRingedSpaceObj_presheaf' (R : Type u) [CommRing R] :
(Spec.locallyRingedSpaceObj <| CommRingCat.of R).presheaf = (structureSheaf R).1 := rfl | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.locallyRingedSpaceObj_presheaf' | null |
Spec.locallyRingedSpaceObj_presheaf_map' (R : Type u) [CommRing R] {U V} (i : U ⟶ V) :
(Spec.locallyRingedSpaceObj <| CommRingCat.of R).presheaf.map i =
(structureSheaf R).1.map i := rfl
@[elementwise] | lemma | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.locallyRingedSpaceObj_presheaf_map' | null |
stalkMap_toStalk {R S : CommRingCat.{u}} (f : R ⟶ S) (p : PrimeSpectrum S) :
toStalk R (PrimeSpectrum.comap f.hom p) ≫ (Spec.sheafedSpaceMap f).stalkMap p =
f ≫ toStalk S p := by
rw [← toOpen_germ S ⊤ p trivial, ← toOpen_germ R ⊤ (PrimeSpectrum.comap f.hom p) trivial,
Category.assoc]
erw [PresheafedSpace.stalkMap_germ (Spec.sheafedSpaceMap f) ⊤ p trivial]
rw [Spec.sheafedSpaceMap_c_app]
erw [toOpen_comp_comap_assoc]
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | stalkMap_toStalk | null |
@[elementwise]
localRingHom_comp_stalkIso {R S : CommRingCat.{u}} (f : R ⟶ S) (p : PrimeSpectrum S) :
(stalkIso R (PrimeSpectrum.comap f.hom p)).hom ≫
(CommRingCat.ofHom (Localization.localRingHom (PrimeSpectrum.comap f.hom p).asIdeal p.asIdeal
f.hom rfl)) ≫
(stalkIso S p).inv =
(Spec.sheafedSpaceMap f).stalkMap p :=
(stalkIso R (PrimeSpectrum.comap f.hom p)).eq_inv_comp.mp <|
(stalkIso S p).comp_inv_eq.mpr <| CommRingCat.hom_ext <|
Localization.localRingHom_unique _ _ _ (PrimeSpectrum.comap_asIdeal _ _) fun x => by
rw [stalkIso_hom, stalkIso_inv, CommRingCat.comp_apply, CommRingCat.comp_apply,
localizationToStalk_of, stalkMap_toStalk_apply f p x]
erw [stalkToFiberRingHom_toStalk]
rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | localRingHom_comp_stalkIso | Under the isomorphisms `stalkIso`, the map `stalkMap (Spec.sheafedSpaceMap f) p` corresponds
to the induced local ring homomorphism `Localization.localRingHom`. |
localRingHom_comp_stalkIso_apply' {R S : CommRingCat.{u}} (f : R ⟶ S) (p : PrimeSpectrum S)
(x) :
(stalkIso S p).inv ((Localization.localRingHom (PrimeSpectrum.comap f.hom p).asIdeal p.asIdeal
f.hom rfl) ((stalkIso R (PrimeSpectrum.comap f.hom p)).hom x)) =
(Spec.sheafedSpaceMap f).stalkMap p x :=
localRingHom_comp_stalkIso_apply _ _ _ | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | localRingHom_comp_stalkIso_apply' | Version of `localRingHom_comp_stalkIso_apply` using `CommRingCat.Hom.hom` |
@[simps toShHom]
Spec.locallyRingedSpaceMap {R S : CommRingCat.{u}} (f : R ⟶ S) :
Spec.locallyRingedSpaceObj S ⟶ Spec.locallyRingedSpaceObj R :=
LocallyRingedSpace.Hom.mk (Spec.sheafedSpaceMap f) fun p =>
IsLocalHom.mk fun a ha => by
erw [← localRingHom_comp_stalkIso_apply' f p a] at ha
have : IsLocalHom (stalkIso (↑S) p).inv.hom := isLocalHom_of_isIso _
replace ha := (isUnit_map_iff (stalkIso S p).inv.hom _).mp ha
replace ha := IsLocalHom.map_nonunit
((stalkIso R ((PrimeSpectrum.comap f.hom) p)).hom a) ha
convert RingHom.isUnit_map (stalkIso R (PrimeSpectrum.comap f.hom p)).inv.hom ha
rw [← CommRingCat.comp_apply, Iso.hom_inv_id, CommRingCat.id_apply]
@[simp] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.locallyRingedSpaceMap | The induced map of a ring homomorphism on the prime spectra, as a morphism of locally ringed spaces. |
Spec.locallyRingedSpaceMap_id (R : CommRingCat.{u}) :
Spec.locallyRingedSpaceMap (𝟙 R) = 𝟙 (Spec.locallyRingedSpaceObj R) :=
LocallyRingedSpace.Hom.ext' <| by
rw [Spec.locallyRingedSpaceMap_toShHom, Spec.sheafedSpaceMap_id]; rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.locallyRingedSpaceMap_id | null |
Spec.locallyRingedSpaceMap_comp {R S T : CommRingCat.{u}} (f : R ⟶ S) (g : S ⟶ T) :
Spec.locallyRingedSpaceMap (f ≫ g) =
Spec.locallyRingedSpaceMap g ≫ Spec.locallyRingedSpaceMap f :=
LocallyRingedSpace.Hom.ext' <| by
rw [Spec.locallyRingedSpaceMap_toShHom, Spec.sheafedSpaceMap_comp]; rfl | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.locallyRingedSpaceMap_comp | null |
@[simps]
Spec.toLocallyRingedSpace : CommRingCat.{u}ᵒᵖ ⥤ LocallyRingedSpace where
obj R := Spec.locallyRingedSpaceObj (unop R)
map f := Spec.locallyRingedSpaceMap f.unop
map_id R := by dsimp; rw [Spec.locallyRingedSpaceMap_id]
map_comp f g := by dsimp; rw [Spec.locallyRingedSpaceMap_comp] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec.toLocallyRingedSpace | Spec, as a contravariant functor from commutative rings to locally ringed spaces. |
toSpecΓ (R : CommRingCat.{u}) : R ⟶ Γ.obj (op (Spec.toLocallyRingedSpace.obj (op R))) :=
StructureSheaf.toOpen R ⊤ | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | toSpecΓ | The counit morphism `R ⟶ Γ(Spec R)` given by `AlgebraicGeometry.StructureSheaf.toOpen`. |
isIso_toSpecΓ (R : CommRingCat.{u}) : IsIso (toSpecΓ R) := by
cases R; apply StructureSheaf.isIso_to_global
@[reassoc] | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | isIso_toSpecΓ | null |
Spec_Γ_naturality {R S : CommRingCat.{u}} (f : R ⟶ S) :
f ≫ toSpecΓ S = toSpecΓ R ≫ Γ.map (Spec.toLocallyRingedSpace.map f.op).op := by
ext : 2
refine Subtype.ext <| funext fun x' => ?_; symm
apply Localization.localRingHom_to_map | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec_Γ_naturality | null |
@[simps! hom_app inv_app]
LocallyRingedSpace.SpecΓIdentity : Spec.toLocallyRingedSpace.rightOp ⋙ Γ ≅ 𝟭 _ :=
Iso.symm <| NatIso.ofComponents.{u,u,u+1,u+1} (fun R =>
letI : IsIso (toSpecΓ R) := StructureSheaf.isIso_to_global _
asIso (toSpecΓ R)) fun {X Y} f => by convert Spec_Γ_naturality (R := X) (S := Y) f | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | LocallyRingedSpace.SpecΓIdentity | The counit (`SpecΓIdentity.inv.op`) of the adjunction `Γ ⊣ Spec` is an isomorphism. |
Spec_map_localization_isIso (R : CommRingCat.{u}) (M : Submonoid R)
(x : PrimeSpectrum (Localization M)) :
IsIso
((Spec.toPresheafedSpace.map
(CommRingCat.ofHom (algebraMap R (Localization M))).op).stalkMap x) := by
dsimp only [Spec.toPresheafedSpace_map, Quiver.Hom.unop_op]
rw [← localRingHom_comp_stalkIso]
refine IsIso.comp_isIso' inferInstance (IsIso.comp_isIso' ?_ inferInstance)
/- I do not know why this is defeq to the goal, but I'm happy to accept that it is. -/
change
IsIso (IsLocalization.localizationLocalizationAtPrimeIsoLocalization M
x.asIdeal).toRingEquiv.toCommRingCatIso.hom
infer_instance | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | Spec_map_localization_isIso | The stalk map of `Spec M⁻¹R ⟶ Spec R` is an iso for each `p : Spec M⁻¹R`. |
toPushforwardStalk : S ⟶ (Spec.topMap f _* (structureSheaf S).1).stalk p :=
StructureSheaf.toOpen S ⊤ ≫
@TopCat.Presheaf.germ _ _ _ _ (Spec.topMap f _* (structureSheaf S).1) ⊤ p trivial
@[reassoc] | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | toPushforwardStalk | For an algebra `f : R →+* S`, this is the ring homomorphism `S →+* (f∗ 𝒪ₛ)ₚ` for a `p : Spec R`.
This is shown to be the localization at `p` in `isLocalizedModule_toPushforwardStalkAlgHom`. |
toPushforwardStalk_comp :
f ≫ StructureSheaf.toPushforwardStalk f p =
StructureSheaf.toStalk R p ≫
(TopCat.Presheaf.stalkFunctor _ _).map (Spec.sheafedSpaceMap f).c := by
rw [StructureSheaf.toStalk, Category.assoc, TopCat.Presheaf.stalkFunctor_map_germ]
exact Spec_Γ_naturality_assoc f _ | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | toPushforwardStalk_comp | null |
algebraMap_pushforward_stalk :
algebraMap R ((Spec.topMap f _* (structureSheaf S).1).stalk p) =
(f ≫ StructureSheaf.toPushforwardStalk f p).hom :=
rfl
variable (R S)
variable [Algebra R S] | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | algebraMap_pushforward_stalk | null |
@[simps!]
toPushforwardStalkAlgHom :
S →ₐ[R] (Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _* (structureSheaf S).1).stalk p :=
{ (StructureSheaf.toPushforwardStalk (CommRingCat.ofHom (algebraMap R S)) p).hom with
commutes' := fun _ => rfl } | def | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | toPushforwardStalkAlgHom | This is the `AlgHom` version of `toPushforwardStalk`, which is the map `S ⟶ (f∗ 𝒪ₛ)ₚ` for some
algebra `R ⟶ S` and some `p : Spec R`. |
isLocalizedModule_toPushforwardStalkAlgHom_aux (y) :
∃ x : S × p.asIdeal.primeCompl, x.2 • y = toPushforwardStalkAlgHom R S p x.1 := by
obtain ⟨U, hp, s, e⟩ := TopCat.Presheaf.germ_exist _ _ y
obtain ⟨_, ⟨r, rfl⟩, hpr : p ∈ PrimeSpectrum.basicOpen r, hrU : PrimeSpectrum.basicOpen r ≤ U⟩ :=
PrimeSpectrum.isTopologicalBasis_basic_opens.exists_subset_of_mem_open (show p ∈ U from hp) U.2
change PrimeSpectrum.basicOpen r ≤ U at hrU
replace e :=
((Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _* (structureSheaf S).1).germ_res_apply
(homOfLE hrU) p hpr _).trans e
set s' := (Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _* (structureSheaf S).1).map
(homOfLE hrU).op s with h
replace e : ((Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _* (structureSheaf S).val).germ _
p hpr) s' = y := by
rw [h]; exact e
clear_value s'; clear! U
obtain ⟨⟨s, ⟨_, n, rfl⟩⟩, hsn⟩ :=
@IsLocalization.surj _ _ _ _ _ _
(StructureSheaf.IsLocalization.to_basicOpen S <| algebraMap R S r) s'
refine ⟨⟨s, ⟨r, hpr⟩ ^ n⟩, ?_⟩
rw [Submonoid.smul_def, Algebra.smul_def, algebraMap_pushforward_stalk, toPushforwardStalk,
CommRingCat.comp_apply, CommRingCat.comp_apply]
iterate 2
erw [← (Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _* (structureSheaf S).1).germ_res_apply
(homOfLE le_top) p hpr]
rw [← e]
let f := TopCat.Presheaf.germ (Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _*
(structureSheaf S).val) _ p hpr
rw [← map_mul, mul_comm]
dsimp only [Subtype.coe_mk] at hsn
rw [← map_pow (algebraMap R S)] at hsn
congr 1 | theorem | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | isLocalizedModule_toPushforwardStalkAlgHom_aux | null |
isLocalizedModule_toPushforwardStalkAlgHom :
IsLocalizedModule p.asIdeal.primeCompl (toPushforwardStalkAlgHom R S p).toLinearMap := by
apply IsLocalizedModule.mkOfAlgebra
· intro x hx; rw [algebraMap_pushforward_stalk, toPushforwardStalk_comp]
change IsUnit ((TopCat.Presheaf.stalkFunctor CommRingCat p).map
(Spec.sheafedSpaceMap (CommRingCat.ofHom (algebraMap ↑R ↑S))).c _)
exact (IsLocalization.map_units ((structureSheaf R).presheaf.stalk p) ⟨x, hx⟩).map _
· apply isLocalizedModule_toPushforwardStalkAlgHom_aux
· intro x hx
rw [toPushforwardStalkAlgHom_apply,
← (toPushforwardStalk (CommRingCat.ofHom (algebraMap ↑R ↑S)) p).hom.map_zero,
toPushforwardStalk] at hx
rw [CommRingCat.comp_apply, map_zero] at hx
obtain ⟨U, hpU, i₁, i₂, e⟩ := TopCat.Presheaf.germ_eq (C := CommRingCat) _ _ _ _ _ _ hx
obtain ⟨_, ⟨r, rfl⟩, hpr, hrU⟩ :=
PrimeSpectrum.isTopologicalBasis_basic_opens.exists_subset_of_mem_open (show p ∈ U.1 from hpU)
U.2
apply_fun (Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _* (structureSheaf S).1).map
(homOfLE hrU).op at e
simp only [map_zero] at e
have : toOpen S (PrimeSpectrum.basicOpen <| algebraMap R S r) x = 0 := by
refine Eq.trans ?_ e; rfl
have :=
(@IsLocalization.mk'_one _ _ _ _ _ _
(StructureSheaf.IsLocalization.to_basicOpen S <| algebraMap R S r) x).trans
this
obtain ⟨⟨_, n, rfl⟩, e⟩ := (IsLocalization.mk'_eq_zero_iff _ _).mp this
refine ⟨⟨r, hpr⟩ ^ n, ?_⟩
rw [Submonoid.smul_def, Algebra.smul_def, SubmonoidClass.coe_pow, map_pow]
exact e | instance | AlgebraicGeometry | [
"Mathlib.Geometry.RingedSpace.LocallyRingedSpace",
"Mathlib.AlgebraicGeometry.StructureSheaf",
"Mathlib.RingTheory.Localization.LocalizationLocalization",
"Mathlib.Topology.Sheaves.SheafCondition.Sites",
"Mathlib.Topology.Sheaves.Functors",
"Mathlib.Algebra.Module.LocalizedModule.Basic"
] | Mathlib/AlgebraicGeometry/Spec.lean | isLocalizedModule_toPushforwardStalkAlgHom | null |
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