mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion

Goal state before:
pp: case inr
X Y : Scheme
inst✝ : Subsingleton ↥Y
f : X ⟶ Y
g : Y ⟶ X
hg : g ≫ f = 𝟙 Y
this : ∀ {X Y : Scheme} [Subsingleton ↥Y] (f : X ⟶ Y) (g : Y ⟶ X), g ≫ f = 𝟙 Y → (∃ R, X = Spec R) → IsClosedImmersion g
hX : ¬∃ R, X = Spec R
⊢ IsClosedImmersion g

pp: X Y : Scheme
inst✝ : Subsingleton ↥Y
f : X ⟶ Y
g : Y ⟶ X
hg : g ≫ f = 𝟙 Y
hX : ∃ R, X = Spec R
⊢ IsClosedImmersion g

Tactic:
·

Goal state after:
case inr
X Y : Scheme
inst✝ : Subsingleton ↥Y
f : X ⟶ Y
g : Y ⟶ X
hg : g ≫ f = 𝟙 Y
this : ∀ {X Y : Scheme} [Subsingleton ↥Y] (f : X ⟶ Y) (g : Y ⟶ X), g ≫ f = 𝟙 Y → (∃ R, X = Spec R) → IsClosedImmersion g
hX : ¬∃ R, X = Spec R
⊢ IsClosedImmersion g
Module: Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion

Goal state before:
pp: case pos
X Y : Scheme
inst✝ : Subsingleton ↥Y
f : X ⟶ Y
g : Y ⟶ X
hg : g ≫ f = 𝟙 Y
this : ∀ {X Y : Scheme} [Subsingleton ↥Y] (f : X ⟶ Y) (g : Y ⟶ X), g ≫ f = 𝟙 Y → (∃ R, X = Spec R) → IsClosedImmersion g
hX : ¬∃ R, X = Spec R
i : X.affineCover.toPreZeroHypercover.1
hxU : Set.range ⇑g ⊆ ↑(Scheme.Hom.opensRange (X.affineCover.f i))
⊢ IsClosedImmersion (Scheme.Cover.pullbackHom X.affineCover g i)

pp: case neg
X Y : Scheme
inst✝ : Subsingleton ↥Y
f : X ⟶ Y
g : Y ⟶ X
hg : g ≫ f = 𝟙 Y
this : ∀ {X Y : Scheme} [Subsingleton ↥Y] (f : X ⟶ Y) (g : Y ⟶ X), g ≫ f = 𝟙 Y → (∃ R, X = Spec R) → IsClosedImmersion g
hX : ¬∃ R, X = Spec R
i : X.affineCover.toPreZeroHypercover.1
hxU : ¬Set.range ⇑g ⊆ ↑(Scheme.Hom.opensRange (X.affineCover.f i))
⊢ IsClosedImmersion (Scheme.Cover.pullbackHom X.affineCover g i)

Tactic:
·

Goal state after:
case pos
X Y : Scheme
inst✝ : Subsingleton ↥Y
f : X ⟶ Y
g : Y ⟶ X
hg : g ≫ f = 𝟙 Y
this : ∀ {X Y : Scheme} [Subsingleton ↥Y] (f : X ⟶ Y) (g : Y ⟶ X), g ≫ f = 𝟙 Y → (∃ R, X = Spec R) → IsClosedImmersion g
hX : ¬∃ R, X = Spec R
i : X.affineCover.toPreZeroHypercover.1
hxU : Set.range ⇑g ⊆ ↑(Scheme.Hom.opensRange (X.affineCover.f i))
⊢ IsClosedImmersion (Scheme.Cover.pullbackHom X.affineCover g i)
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: case inl
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Subsingleton R
⊢ IsNilpotent ((algebraMap A (A ⊗[R] I.ResidueField)) f) ↔
    ∀ i < Module.finrank R A, (LinearMap.charpoly ((Algebra.lmul R A) f)).coeff i ∈ I

pp: case inr
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
⊢ IsNilpotent ((algebraMap A (A ⊗[R] I.ResidueField)) f) ↔
    ∀ i < Module.finrank R A, (LinearMap.charpoly ((Algebra.lmul R A) f)).coeff i ∈ I

Tactic:
·

Goal state after:
case inl
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Subsingleton R
⊢ IsNilpotent ((algebraMap A (A ⊗[R] I.ResidueField)) f) ↔
    ∀ i < Module.finrank R A, (LinearMap.charpoly ((Algebra.lmul R A) f)).coeff i ∈ I
Module: Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion

Goal state before:
pp: case neg
X Y : Scheme
inst✝ : Subsingleton ↥Y
f : X ⟶ Y
g : Y ⟶ X
hg : g ≫ f = 𝟙 Y
this : ∀ {X Y : Scheme} [Subsingleton ↥Y] (f : X ⟶ Y) (g : Y ⟶ X), g ≫ f = 𝟙 Y → (∃ R, X = Spec R) → IsClosedImmersion g
hX : ¬∃ R, X = Spec R
i : X.affineCover.toPreZeroHypercover.1
hxU : ¬Set.range ⇑g ⊆ ↑(Scheme.Hom.opensRange (X.affineCover.f i))
⊢ IsClosedImmersion (Scheme.Cover.pullbackHom X.affineCover g i)

Tactic:
·

Goal state after:
case neg
X Y : Scheme
inst✝ : Subsingleton ↥Y
f : X ⟶ Y
g : Y ⟶ X
hg : g ≫ f = 𝟙 Y
this : ∀ {X Y : Scheme} [Subsingleton ↥Y] (f : X ⟶ Y) (g : Y ⟶ X), g ≫ f = 𝟙 Y → (∃ R, X = Spec R) → IsClosedImmersion g
hX : ¬∃ R, X = Spec R
i : X.affineCover.toPreZeroHypercover.1
hxU : ¬Set.range ⇑g ⊆ ↑(Scheme.Hom.opensRange (X.affineCover.f i))
⊢ IsClosedImmersion (Scheme.Cover.pullbackHom X.affineCover g i)
Module: Mathlib.RingTheory.Spectrum.Prime.Chevalley

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FinitePresentation
s : Set (PrimeSpectrum S)
hs : IsConstructible s
⊢ ∀ (R : Type u_1) [inst : CommRing R] (S : Type u_2) [inst_1 : CommRing S] (f : R →+* S),
    Surjective ⇑f →
      (RingHom.ker f).FG → ∀ (s : Set (PrimeSpectrum S)), IsConstructible s → IsConstructible (comap f '' s)

pp: case refine_2
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FinitePresentation
s : Set (PrimeSpectrum S)
hs : IsConstructible s
⊢ ∀ (R : Type u_1) [inst : CommRing R] (S : Type u_1) [inst_1 : CommRing S] (T : Type u_2) [inst_2 : CommRing T]
    (f : R →+* S) (g : S →+* T),
    (∀ (s : Set (PrimeSpectrum S)), IsConstructible s → IsConstructible (comap f '' s)) →
      (∀ (s : Set (PrimeSpectrum T)), IsConstructible s → IsConstructible (comap g '' s)) →
        ∀ (s : Set (PrimeSpectrum T)), IsConstructible s → IsConstructible (comap (g.comp f) '' s)

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FinitePresentation
s : Set (PrimeSpectrum S)
hs : IsConstructible s
⊢ ∀ (R : Type u_1) [inst : CommRing R] (S : Type u_2) [inst_1 : CommRing S] (f : R →+* S),
    Surjective ⇑f →
      (RingHom.ker f).FG → ∀ (s : Set (PrimeSpectrum S)), IsConstructible s → IsConstructible (comap f '' s)
Module: Mathlib.RingTheory.Spectrum.Prime.Chevalley

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FinitePresentation
s : Set (PrimeSpectrum S)
hs : IsConstructible s
⊢ ∀ (R : Type u_1) [inst : CommRing R] (S : Type u_1) [inst_1 : CommRing S] (T : Type u_2) [inst_2 : CommRing T]
    (f : R →+* S) (g : S →+* T),
    (∀ (s : Set (PrimeSpectrum S)), IsConstructible s → IsConstructible (comap f '' s)) →
      (∀ (s : Set (PrimeSpectrum T)), IsConstructible s → IsConstructible (comap g '' s)) →
        ∀ (s : Set (PrimeSpectrum T)), IsConstructible s → IsConstructible (comap (g.comp f) '' s)

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.FinitePresentation
s : Set (PrimeSpectrum S)
hs : IsConstructible s
⊢ ∀ (R : Type u_1) [inst : CommRing R] (S : Type u_1) [inst_1 : CommRing S] (T : Type u_2) [inst_2 : CommRing T]
    (f : R →+* S) (g : S →+* T),
    (∀ (s : Set (PrimeSpectrum S)), IsConstructible s → IsConstructible (comap f '' s)) →
      (∀ (s : Set (PrimeSpectrum T)), IsConstructible s → IsConstructible (comap g '' s)) →
        ∀ (s : Set (PrimeSpectrum T)), IsConstructible s → IsConstructible (comap (g.comp f) '' s)
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: case inr.mp
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
⊢ Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f)) =
      X ^ ((LinearMap.mul R A) f).charpoly.natDegree →
    ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I

pp: case inr.mpr
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
⊢ (∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I) →
    Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f)) =
      X ^ ((LinearMap.mul R A) f).charpoly.natDegree

Tactic:
·

Goal state after:
case inr.mp
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
⊢ Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f)) =
      X ^ ((LinearMap.mul R A) f).charpoly.natDegree →
    ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I
Module: Mathlib.RingTheory.Spectrum.Prime.Chevalley

Goal state before:
pp: case inr
K : Type u_3
A : Type u_4
B : Type u_5
inst✝⁴ : Field K
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra K A
inst✝ : Algebra K B
U : Set (PrimeSpectrum (A ⊗[K] B))
hU : IsOpen U
this :
  ∀ (U : Set (PrimeSpectrum (A ⊗[K] B))),
    IsOpen U → (∃ f, U = ↑(basicOpen f)) → IsOpen (comap (algebraMap A (A ⊗[K] B)) '' U)
hU' : ¬∃ f, U = ↑(basicOpen f)
⊢ IsOpen (comap (algebraMap A (A ⊗[K] B)) '' U)

pp: K : Type u_3
A : Type u_4
B : Type u_5
inst✝⁴ : Field K
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra K A
inst✝ : Algebra K B
U : Set (PrimeSpectrum (A ⊗[K] B))
hU : IsOpen U
hU' : ∃ f, U = ↑(basicOpen f)
⊢ IsOpen (comap (algebraMap A (A ⊗[K] B)) '' U)

Tactic:
·

Goal state after:
case inr
K : Type u_3
A : Type u_4
B : Type u_5
inst✝⁴ : Field K
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra K A
inst✝ : Algebra K B
U : Set (PrimeSpectrum (A ⊗[K] B))
hU : IsOpen U
this :
  ∀ (U : Set (PrimeSpectrum (A ⊗[K] B))),
    IsOpen U → (∃ f, U = ↑(basicOpen f)) → IsOpen (comap (algebraMap A (A ⊗[K] B)) '' U)
hU' : ¬∃ f, U = ↑(basicOpen f)
⊢ IsOpen (comap (algebraMap A (A ⊗[K] B)) '' U)
Module: Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion

Goal state before:
pp: case inr
Y : Scheme
inst✝ : Subsingleton ↥Y
R : CommRingCat
f : Spec R ⟶ Y
g : Y ⟶ Spec R
hg : g ≫ f = 𝟙 Y
this :
  ∀ {Y : Scheme} [Subsingleton ↥Y] (R : CommRingCat) (f : Spec R ⟶ Y) (g : Y ⟶ Spec R),
    g ≫ f = 𝟙 Y → (∃ S, Y = Spec S) → IsClosedImmersion g
hY : ¬∃ S, Y = Spec S
⊢ IsClosedImmersion g

pp: Y : Scheme
inst✝ : Subsingleton ↥Y
R : CommRingCat
f : Spec R ⟶ Y
g : Y ⟶ Spec R
hg : g ≫ f = 𝟙 Y
hY : ∃ S, Y = Spec S
⊢ IsClosedImmersion g

Tactic:
·

Goal state after:
case inr
Y : Scheme
inst✝ : Subsingleton ↥Y
R : CommRingCat
f : Spec R ⟶ Y
g : Y ⟶ Spec R
hg : g ≫ f = 𝟙 Y
this :
  ∀ {Y : Scheme} [Subsingleton ↥Y] (R : CommRingCat) (f : Spec R ⟶ Y) (g : Y ⟶ Spec R),
    g ≫ f = 𝟙 Y → (∃ S, Y = Spec S) → IsClosedImmersion g
hY : ¬∃ S, Y = Spec S
⊢ IsClosedImmersion g
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: case inr.mpr
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
⊢ (∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I) →
    Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f)) =
      X ^ ((LinearMap.mul R A) f).charpoly.natDegree

Tactic:
·

Goal state after:
case inr.mpr
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
⊢ (∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I) →
    Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f)) =
      X ^ ((LinearMap.mul R A) f).charpoly.natDegree
Module: Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion

Goal state before:
pp: R S : CommRingCat
inst✝ : Subsingleton ↥(Spec S)
φ : S ⟶ R
ψ : R ⟶ S
hg : Spec.map (φ ≫ ψ) = 𝟙 (Spec S)
⊢ IsClosedImmersion (Spec.map ψ)

Tactic:
← Spec.map_id,

Goal state after:
R S : CommRingCat
inst✝ : Subsingleton ↥(Spec S)
φ : S ⟶ R
ψ : R ⟶ S
hg : Spec.map (φ ≫ ψ) = Spec.map (𝟙 S)
⊢ IsClosedImmersion (Spec.map ψ)
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: case inr.mpr.inl
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
H : ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I
i : ℕ
hi : i = ((LinearMap.mul R A) f).charpoly.natDegree
⊢ (Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f))).coeff i =
    (X ^ ((LinearMap.mul R A) f).charpoly.natDegree).coeff i

pp: case inr.mpr.inr
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
H : ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I
i : ℕ
hi : i ≠ ((LinearMap.mul R A) f).charpoly.natDegree
⊢ (Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f))).coeff i =
    (X ^ ((LinearMap.mul R A) f).charpoly.natDegree).coeff i

Tactic:
·

Goal state after:
case inr.mpr.inl
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
H : ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I
i : ℕ
hi : i = ((LinearMap.mul R A) f).charpoly.natDegree
⊢ (Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f))).coeff i =
    (X ^ ((LinearMap.mul R A) f).charpoly.natDegree).coeff i
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: case inr.mpr.inr.inl
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
H : ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I
i : ℕ
hi✝ : i ≠ ((LinearMap.mul R A) f).charpoly.natDegree
hi : i < ((LinearMap.mul R A) f).charpoly.natDegree
⊢ (Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f))).coeff i =
    (X ^ ((LinearMap.mul R A) f).charpoly.natDegree).coeff i

pp: case inr.mpr.inr.inr
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
H : ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I
i : ℕ
hi✝ : i ≠ ((LinearMap.mul R A) f).charpoly.natDegree
hi : ((LinearMap.mul R A) f).charpoly.natDegree < i
⊢ (Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f))).coeff i =
    (X ^ ((LinearMap.mul R A) f).charpoly.natDegree).coeff i

Tactic:
·

Goal state after:
case inr.mpr.inr.inl
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
H : ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I
i : ℕ
hi✝ : i ≠ ((LinearMap.mul R A) f).charpoly.natDegree
hi : i < ((LinearMap.mul R A) f).charpoly.natDegree
⊢ (Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f))).coeff i =
    (X ^ ((LinearMap.mul R A) f).charpoly.natDegree).coeff i
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: case inr.mpr.inr.inr
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
H : ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I
i : ℕ
hi✝ : i ≠ ((LinearMap.mul R A) f).charpoly.natDegree
hi : ((LinearMap.mul R A) f).charpoly.natDegree < i
⊢ (Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f))).coeff i =
    (X ^ ((LinearMap.mul R A) f).charpoly.natDegree).coeff i

Tactic:
·

Goal state after:
case inr.mpr.inr.inr
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Module.Free R A
inst✝¹ : Module.Finite R A
f : A
I : Ideal R
inst✝ : I.IsPrime
h✝ : Nontrivial R
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
H : ∀ i < ((LinearMap.mul R A) f).charpoly.natDegree, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I
i : ℕ
hi✝ : i ≠ ((LinearMap.mul R A) f).charpoly.natDegree
hi : ((LinearMap.mul R A) f).charpoly.natDegree < i
⊢ (Polynomial.map (algebraMap R I.ResidueField) (LinearMap.charpoly ((Algebra.lmul R A) f))).coeff i =
    (X ^ ((LinearMap.mul R A) f).charpoly.natDegree).coeff i
Module: Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation

Goal state before:
pp: case inr
X Y : Scheme
f : X ⟶ Y
hf : LocallyOfFinitePresentation f
inst✝ : QuasiCompact f
s : Set ↥X
hs : IsLocallyConstructible s
this :
  ∀ {X Y : Scheme} (f : X ⟶ Y) [hf : LocallyOfFinitePresentation f] [QuasiCompact f] {s : Set ↥X},
    IsLocallyConstructible s → (∃ R, Y = Spec R) → IsLocallyConstructible (⇑f '' s)
hY : ¬∃ R, Y = Spec R
⊢ IsLocallyConstructible (⇑f '' s)

pp: X Y : Scheme
f : X ⟶ Y
hf : LocallyOfFinitePresentation f
inst✝ : QuasiCompact f
s : Set ↥X
hs : IsLocallyConstructible s
hY : ∃ R, Y = Spec R
⊢ IsLocallyConstructible (⇑f '' s)

Tactic:
·

Goal state after:
case inr
X Y : Scheme
f : X ⟶ Y
hf : LocallyOfFinitePresentation f
inst✝ : QuasiCompact f
s : Set ↥X
hs : IsLocallyConstructible s
this :
  ∀ {X Y : Scheme} (f : X ⟶ Y) [hf : LocallyOfFinitePresentation f] [QuasiCompact f] {s : Set ↥X},
    IsLocallyConstructible s → (∃ R, Y = Spec R) → IsLocallyConstructible (⇑f '' s)
hY : ¬∃ R, Y = Spec R
⊢ IsLocallyConstructible (⇑f '' s)
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: case mp
R : Type u_2
A : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
s : Set A
x : PrimeSpectrum R
⊢ x ∈ comap (algebraMap R A) '' (zeroLocus s \ zeroLocus {f}) →
    ¬IsNilpotent ((algebraMap A ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)) f)

pp: case mpr
R : Type u_2
A : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
s : Set A
x : PrimeSpectrum R
⊢ ¬IsNilpotent ((algebraMap A ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)) f) →
    x ∈ comap (algebraMap R A) '' (zeroLocus s \ zeroLocus {f})

Tactic:
·

Goal state after:
case mp
R : Type u_2
A : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
s : Set A
x : PrimeSpectrum R
⊢ x ∈ comap (algebraMap R A) '' (zeroLocus s \ zeroLocus {f}) →
    ¬IsNilpotent ((algebraMap A ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)) f)
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: case mpr
R : Type u_2
A : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
s : Set A
x : PrimeSpectrum R
⊢ ¬IsNilpotent ((algebraMap A ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)) f) →
    x ∈ comap (algebraMap R A) '' (zeroLocus s \ zeroLocus {f})

Tactic:
·

Goal state after:
case mpr
R : Type u_2
A : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
s : Set A
x : PrimeSpectrum R
⊢ ¬IsNilpotent ((algebraMap A ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)) f) →
    x ∈ comap (algebraMap R A) '' (zeroLocus s \ zeroLocus {f})
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
x : PrimeSpectrum R
⊢ x ∈ comap C '' ↑(basicOpen f) ↔ map (algebraMap R x.asIdeal.ResidueField) f ≠ 0

pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
x : PrimeSpectrum R
⊢ map (algebraMap R x.asIdeal.ResidueField) f ≠ 0 ↔ ∃ i, f.coeff i ∉ x.asIdeal

Tactic:
·

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R[X]
x : PrimeSpectrum R
⊢ x ∈ comap C '' ↑(basicOpen f) ↔ map (algebraMap R x.asIdeal.ResidueField) f ≠ 0
Module: Mathlib.AlgebraicGeometry.Noetherian

Goal state before:
pp: case mp
X : Scheme
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
⊢ IsLocallyNoetherian X → ∀ (i : 𝒰.I₀), IsNoetherianRing ↑Γ(𝒰.X i, ⊤)

pp: case mpr
X : Scheme
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
⊢ (∀ (i : 𝒰.I₀), IsNoetherianRing ↑Γ(𝒰.X i, ⊤)) → IsLocallyNoetherian X

Tactic:
·

Goal state after:
case mp
X : Scheme
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
⊢ IsLocallyNoetherian X → ∀ (i : 𝒰.I₀), IsNoetherianRing ↑Γ(𝒰.X i, ⊤)
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
x : PrimeSpectrum R
⊢ map (algebraMap R x.asIdeal.ResidueField) f ≠ 0 ↔ ∃ i, f.coeff i ∉ x.asIdeal

Tactic:
·

Goal state after:
R : Type u_1
inst✝ : CommRing R
f : R[X]
x : PrimeSpectrum R
⊢ map (algebraMap R x.asIdeal.ResidueField) f ≠ 0 ↔ ∃ i, f.coeff i ∉ x.asIdeal
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R[X]
x : PrimeSpectrum R
⊢ map (algebraMap R x.asIdeal.ResidueField) f ≠ 0 ↔ ∃ i, f.coeff i ∉ x.asIdeal

Tactic:
simp [Polynomial.ext_iff]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation

Goal state before:
pp: case inr
X : Scheme
s : Set ↥X
hs : IsLocallyConstructible s
R : CommRingCat
f : X ⟶ Spec R
hf : LocallyOfFinitePresentation f
inst✝ : QuasiCompact f
this :
  ∀ {X : Scheme} {s : Set ↥X},
    IsLocallyConstructible s →
      ∀ (R : CommRingCat) (f : X ⟶ Spec R) [hf : LocallyOfFinitePresentation f] [QuasiCompact f],
        (∃ S, X = Spec S) → IsLocallyConstructible (⇑f '' s)
hX : ¬∃ S, X = Spec S
⊢ IsLocallyConstructible (⇑f '' s)

pp: X : Scheme
s : Set ↥X
hs : IsLocallyConstructible s
R : CommRingCat
f : X ⟶ Spec R
hf : LocallyOfFinitePresentation f
inst✝ : QuasiCompact f
hX : ∃ S, X = Spec S
⊢ IsLocallyConstructible (⇑f '' s)

Tactic:
·

Goal state after:
case inr
X : Scheme
s : Set ↥X
hs : IsLocallyConstructible s
R : CommRingCat
f : X ⟶ Spec R
hf : LocallyOfFinitePresentation f
inst✝ : QuasiCompact f
this :
  ∀ {X : Scheme} {s : Set ↥X},
    IsLocallyConstructible s →
      ∀ (R : CommRingCat) (f : X ⟶ Spec R) [hf : LocallyOfFinitePresentation f] [QuasiCompact f],
        (∃ S, X = Spec S) → IsLocallyConstructible (⇑f '' s)
hX : ¬∃ S, X = Spec S
⊢ IsLocallyConstructible (⇑f '' s)
Module: Mathlib.AlgebraicGeometry.Noetherian

Goal state before:
pp: case mpr
X : Scheme
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
⊢ (∀ (i : 𝒰.I₀), IsNoetherianRing ↑Γ(𝒰.X i, ⊤)) → IsLocallyNoetherian X

Tactic:
·

Goal state after:
case mpr
X : Scheme
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
⊢ (∀ (i : 𝒰.I₀), IsNoetherianRing ↑Γ(𝒰.X i, ⊤)) → IsLocallyNoetherian X
Module: Mathlib.RingTheory.Spectrum.Prime.Polynomial

Goal state before:
pp: case inst
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hg : g.Monic
⊢ Module.Finite R (R[X] ⧸ Ideal.span {g})

pp: case inst
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hg : g.Monic
⊢ Module.Free R (R[X] ⧸ Ideal.span {g})

Tactic:
·

Goal state after:
case inst
R : Type u_1
inst✝ : CommRing R
f g : R[X]
hg : g.Monic
⊢ Module.Finite R (R[X] ⧸ Ideal.span {g})