mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ f ^ i ∈ 𝒜 (m * i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
f g x : A
hx : x = f * g
n : ι
a : ↥(𝒜 n)
i : ℕ
hi : f ^ i ∈ 𝒜 n
⊢ x ^ 1 = f * g

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
f g x : A
hx : x = f * g
n : ι
a : ↥(𝒜 n)
i : ℕ
hi : f ^ i ∈ 𝒜 n
⊢ (algebraMap A (Localization.Away x)) f * Localization.mk g ⟨f * g, ⋯⟩ = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
f g x : A
hx : x = f * g
n : ι
a : ↥(𝒜 n)
i : ℕ
hi : f ^ i ∈ 𝒜 n
⊢ (awayMapAux 𝒜 ⋯) (mk { deg := n, num := a, den := ⟨f ^ i, hi⟩, den_mem := ⋯ }) =
    Localization.mk (↑a * g ^ i) ⟨x ^ i, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
e : ι
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
⊢ Set.range ⇑(awayMapAux 𝒜 ⋯) ⊆ Set.range val

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
e : ι
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
⊢ Away 𝒜 f →+* Away 𝒜 x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
e : ι
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
a : Away 𝒜 f
⊢ val ((awayMap 𝒜 hg hx) a) = (awayMapAux 𝒜 ⋯) a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
e : ι
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
a : Away 𝒜 f
⊢ val ((awayMap 𝒜 hg hx) a) = (Localization.awayLift (algebraMap A (at Submonoid.powers x)) f ⋯) (val a)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
e : ι
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
a : ↥(𝒜 0)
⊢ (awayMap 𝒜 hg hx) ((fromZeroRingHom 𝒜 (Submonoid.powers f)) a) = (fromZeroRingHom 𝒜 (Submonoid.powers x)) a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
e : ι
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
n : ι
a : ↥(𝒜 n)
i : ℕ
hi : f ^ i ∈ 𝒜 n
⊢ val ((awayMap 𝒜 hg hx) (mk { deg := n, num := a, den := ⟨f ^ i, hi⟩, den_mem := ⋯ })) =
    Localization.mk (↑a * g ^ i) ⟨x ^ i, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
ι : Type u_2
f : ι → A
hfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n
hf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤
⊢ ⨆ i, basicOpen 𝒜 (f i) = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
⊢ Scheme.Hom.app (basicOpenToSpec 𝒜 f) ⊤ =
    (Scheme.ΓSpecIso (CommRingCat.of (Away 𝒜 f))).hom ≫ awayToSection 𝒜 f ≫ (basicOpen 𝒜 f).topIso.inv

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
e : ι
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
d : ι
n : ℕ
hf : f ∈ 𝒜 d
a : A
ha : a ∈ 𝒜 (n • d)
⊢ a * g ^ n ∈ 𝒜 (n • (d + e))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f g : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
⊢ IsIso (basicOpenToSpec 𝒜 f)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
e : ι
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
d : ι
n : ℕ
hf : f ∈ 𝒜 d
a : A
ha : a ∈ 𝒜 (n • d)
⊢ (awayMap 𝒜 hg hx) (Away.mk 𝒜 hf n a ha) = Away.mk 𝒜 ⋯ n (a * g ^ n) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
e d : ℕ
f : A
hf : f ∈ 𝒜 d
g : A
hg : g ∈ 𝒜 e
⊢ g ^ d ∈ 𝒜 (e • d)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f g : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
⊢ IsIso (awayToSection 𝒜 f)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
⊢ IsAffineOpen (basicOpen 𝒜 f)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
U : (Proj 𝒜).Opens
e : U = ⊤
⊢ (basicOpen 𝒜 f).ι ≫ (Proj 𝒜).topIso.inv ≫ ((Proj 𝒜).isoOfEq e).inv = (Proj 𝒜).homOfLE ⋯

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
⊢ awayι 𝒜 f f_deg hm ≫ toSpecZero 𝒜 = Spec.map (CommRingCat.ofHom (fromZeroRingHom 𝒜 (Submonoid.powers f)))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: A : Type u_2
σ : Type u_3
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
e d : ℕ
f : A
hf : f ∈ 𝒜 d
g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
hd : d ≠ 0
this : Algebra (Away 𝒜 f) (Away 𝒜 x) := (awayMap 𝒜 hg hx).toAlgebra
n : ℕ
⊢ g ^ e * f ^ (2 * e + d) ∈ 𝒜 ((d + e) • (d + e))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
x : A
hx : x = f * g
⊢ CommRingCat.ofHom (awayMap 𝒜 g_deg hx) ≫ awayToSection 𝒜 x = awayToSection 𝒜 f ≫ (Proj 𝒜).presheaf.map (homOfLE ⋯).op

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
x : A
hx : x = f * g
⊢ basicOpenToSpec 𝒜 x ≫ Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) = (Proj 𝒜).homOfLE ⋯ ≫ basicOpenToSpec 𝒜 f

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
x : A
hx : x = f * g
⊢ Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) ≫ awayι 𝒜 f f_deg hm = awayι 𝒜 x ⋯ ⋯

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f g✝ : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
x : A
hx : x = f * g
⊢ Set.range ⇑(Limits.pullback.fst (awayι 𝒜 f f_deg hm) (awayι 𝒜 g g_deg hm') ≫ awayι 𝒜 f f_deg hm) =
    Set.range ⇑(awayι 𝒜 x ⋯ ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
x : A
hx : x = f * g
⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫ Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) =
    Limits.pullback.fst (awayι 𝒜 f f_deg hm) (awayι 𝒜 g g_deg hm')

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
x : A
hx : x = f * g
⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫ Spec.map (CommRingCat.ofHom (awayMap 𝒜 f_deg ⋯)) =
    Limits.pullback.snd (awayι 𝒜 f f_deg hm) (awayι 𝒜 g g_deg hm')

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
x : A
hx : x = f * g
⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).inv ≫ Limits.pullback.fst (awayι 𝒜 f f_deg hm) (awayι 𝒜 g g_deg hm') =
    Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg hx))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
x : A
hx : x = f * g
⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).inv ≫ Limits.pullback.snd (awayι 𝒜 f f_deg hm) (awayι 𝒜 g g_deg hm') =
    Spec.map (CommRingCat.ofHom (awayMap 𝒜 f_deg ⋯))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
⊢ Function.Surjective ⇑(TopCat.Hom.hom (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' ⋯).inv.base)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: A : Type u_2
σ : Type u_3
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
e : ℕ
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
this : Algebra (Away 𝒜 f) (Away 𝒜 x) := (awayMap 𝒜 hg hx).toAlgebra
n : ℕ
s : A
d : ℕ
hf : f ∈ 𝒜 (d + 1)
hd : d + 1 ≠ 0
hs : s ∈ 𝒜 (n • (d + 1 + e))
⊢ s * g ^ (n * d) ∈ 𝒜 ((n * (e + 1)) • (d + 1))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
⊢ awayι 𝒜 f f_deg hm ⁻¹ᵁ basicOpen 𝒜 g = PrimeSpectrum.basicOpen (Away.isLocalizationElem f_deg g_deg)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f✝ g✝ : A
m✝ : ℕ
f_deg✝ : f✝ ∈ 𝒜 m✝
hm✝ : 0 < m✝
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
x✝ : A
hx : x✝ = f✝ * g
ι : Type u_2
f : ι → A
m : ι → ℕ
f_deg : ∀ (i : ι), f i ∈ 𝒜 (m i)
hm : ∀ (i : ι), 0 < m i
hf : (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ Ideal.span (Set.range f)
x : ↥(Proj 𝒜)
⊢ x ∈ Set.range ⇑(awayι 𝒜 (f ⋯.choose) ⋯ ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f g✝ : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
x : A
hx : x = f * g
⊢ (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ Ideal.span (Set.range fun i ↦ ↑i.snd)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
H : f t = x
h0d : 0 < d
hd : t ∈ 𝒜 d
⊢ ↑(X.basicOpen x) ⟶ ↑(basicOpen 𝒜 t)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
H : f t = x
h0d : 0 < d
hd : t ∈ 𝒜 d
H' : f t' = x'
h0d' : 0 < d'
hd' : t' ∈ 𝒜 d'
s : A
hts : t * s = t'
n : ℕ
hn : d + n = d'
hs : s ∈ 𝒜 n
⊢ X.basicOpen x' ≤ X.basicOpen x

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
H : f t = x
h0d : 0 < d
hd : t ∈ 𝒜 d
H' : f t' = x'
h0d' : 0 < d'
hd' : t' ∈ 𝒜 d'
s : A
hts : t * s = t'
n : ℕ
hn : d + n = d'
hs : s ∈ 𝒜 n
⊢ PrimeSpectrum.basicOpen x' ≤ PrimeSpectrum.basicOpen x

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
H : f t = x
h0d : 0 < d
hd : t ∈ 𝒜 d
H' : f t' = x'
h0d' : 0 < d'
hd' : t' ∈ 𝒜 d'
s : A
hts : t * s = t'
n : ℕ
hn : d + n = d'
hs : s ∈ 𝒜 n
hx'x : PrimeSpectrum.basicOpen x' ≤ PrimeSpectrum.basicOpen x
⊢ X.basicOpen x' ≤ X.toSpecΓ ⁻¹ᵁ PrimeSpectrum.basicOpen x'

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t : A
d : ℕ
H : f t = x
h0d : 0 < d
hd : t ∈ 𝒜 d
s : A
n : ℕ
hs : s ∈ 𝒜 n
hx'x : PrimeSpectrum.basicOpen x' ≤ PrimeSpectrum.basicOpen x
H' : f (t * s) = x'
h0d' : 0 < d + n
hd' : t * s ∈ 𝒜 (d + n)
⊢ x' = x * f s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: A : Type u_2
σ : Type u_3
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
e : ℕ
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
this : Algebra (Away 𝒜 f) (Away 𝒜 x) := (awayMap 𝒜 hg hx).toAlgebra
n : ℕ
s : A
d : ℕ
hf : f ∈ 𝒜 (d + 1)
hd : d + 1 ≠ 0
hs : s ∈ 𝒜 (n • (d + 1 + e))
t : Away 𝒜 f := Away.mk 𝒜 hf (n * (e + 1)) (s * g ^ (n * d)) ⋯
⊢ ↑1 *
      (↑(s * g ^ (n * d) * g ^ (n * (e + 1)), ⟨x ^ (n * (e + 1)), ⋯⟩).2 *
        (s * (g ^ (d + 1) * g ^ e) ^ n, ⟨x ^ n, ⋯⟩ * ⟨x ^ e, ⋯⟩ ^ n).1) =
    ↑1 *
      (↑(s * (g ^ (d + 1) * g ^ e) ^ n, ⟨x ^ n, ⋯⟩ * ⟨x ^ e, ⋯⟩ ^ n).2 *
        (s * g ^ (n * d) * g ^ (n * (e + 1)), ⟨x ^ (n * (e + 1)), ⋯⟩).1)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
H : f t = x
h0d : 0 < d
hd : t ∈ 𝒜 d
H' : f t' = x'
h0d' : 0 < d'
hd' : t' ∈ 𝒜 d'
s : A
hts : t * s = t'
n : ℕ
hn : d + n = d'
hs : s ∈ 𝒜 n
⊢ X.homOfLE ⋯ ≫ toBasicOpenOfGlobalSections 𝒜 f H h0d hd ≫ (basicOpen 𝒜 t).ι =
    toBasicOpenOfGlobalSections 𝒜 f H' h0d' hd' ≫ (basicOpen 𝒜 t').ι

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f✝ : A →+* ↑Γ(X, ⊤)
x✝ x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
x : A
hx : x ∈ (HomogeneousIdeal.irrelevant 𝒜).toIdeal
c : ℕ
hc : c ∈ DFinsupp.support ((DirectSum.decompose 𝒜) x)
⊢ c ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f✝ : A →+* ↑Γ(X, ⊤)
x✝ x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
x : A
hx : x ∈ (HomogeneousIdeal.irrelevant 𝒜).toIdeal
c : ℕ
hc : c ∈ DFinsupp.support ((DirectSum.decompose 𝒜) x)
this : c ≠ 0
⊢ (((DirectSum.decompose 𝒜) x) c).1 ∈ 𝒜 c

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f✝ : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
⊢ Ideal.span (Set.range fun x ↦ x.snd.fst) = (HomogeneousIdeal.irrelevant 𝒜).toIdeal

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f✝ : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
⊢ TopologicalSpace.IsOpenCover fun ir ↦ X.basicOpen (f ir.snd.fst)

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f✝ : A →+* ↑Γ(X, ⊤)
x✝ x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
x y : (openCoverOfMapIrrelevantEqTop 𝒜 f hf).I₀
⊢ (fun ir ↦ X.basicOpen (f ir.snd.fst)) x ⊓ (fun ir ↦ X.basicOpen (f ir.snd.fst)) y =
    X.basicOpen (f (x.snd.fst * y.snd.fst))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f✝ : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
⊢ X ⟶ Proj 𝒜

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: A : Type u_2
σ : Type u_3
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
e d : ℕ
f : A
hf : f ∈ 𝒜 d
g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
hd : d ≠ 0
⊢ IsLocalization.Away (isLocalizationElem hf hg) (Away 𝒜 x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
⊢ a ∈ Submodule.span ↥(𝒜 0) ↑(Submonoid.closure (Set.range v))

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M✝ : ModuleCat ↑R
M N : (Spec R).Modules
f : modulesSpecToSheaf.obj M ⟶ modulesSpecToSheaf.obj N
U : (Opens ↥(Spec R))ᵒᵖ
⊢ ∀ (m : ↑((Spec R).ringCatSheaf.obj.obj U)) (x : ↑(M.val.1 U)),
    (ModuleCat.Hom.hom (f.hom.app U)).toFun (m • x) =
      (RingHom.id ↑((Spec R).ringCatSheaf.obj.obj U)) m • (ModuleCat.Hom.hom (f.hom.app U)).toFun x

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M✝ : ModuleCat ↑R
M N : (Spec R).Modules
f : modulesSpecToSheaf.obj M ⟶ modulesSpecToSheaf.obj N
X✝ Y✝ : (Opens ↥(Spec R))ᵒᵖ
i : X✝ ⟶ Y✝
⊢ M.val.map i ≫
      (ModuleCat.restrictScalars (RingCat.Hom.hom ((Spec R).ringCatSheaf.obj.map i))).map
        (ModuleCat.ofHom { toAddHom := (ModuleCat.Hom.hom (f.hom.app Y✝)).toAddHom, map_smul' := ⋯ }) =
    ModuleCat.ofHom { toAddHom := (ModuleCat.Hom.hom (f.hom.app X✝)).toAddHom, map_smul' := ⋯ } ≫ N.val.map i

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
ai : ι' → ℕ
H : ¬∑ i, ai i • dv i = n
⊢ ↑(((decompose 𝒜) (∏ i, v i ^ ai i)) n) = 0

Tactic:
by

Goal state after:
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
U : (Spec R).Opens
f : ↑R
hf : U ≤ PrimeSpectrum.basicOpen f
⊢ IsUnit ((algebraMap (↑R) (Module.End ↑R ↑Γ(M, U))) f)

Tactic:
by

Goal state after: