-- Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean
/-- If `A` is a finite free `R`-algebra, then `f : A` is nilpotent on `κ(𝔭) ⊗ A` for some
prime `𝔭 ◃ R` if and only if every non-leading coefficient of `charpoly(f)` is in `𝔭`. -/
lemma isNilpotent_tensor_residueField_iff
    [Module.Free R A] [Module.Finite R A] (f : A) (I : Ideal R) [I.IsPrime] :
    IsNilpotent (algebraMap A (A ⊗[R] I.ResidueField) f) ↔
      ∀ i < Module.finrank R A, (Algebra.lmul R A f).charpoly.coeff i ∈ I

Proof state:
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:
intro H

Result:
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
H : ∀ 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

Proof state:
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

Tactic:
have := (algebraMap R (A ⊗[R] I.ResidueField)).codomain_trivial

Result:
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
this : Subsingleton (A ⊗[R] I.ResidueField)
⊢ IsNilpotent ((algebraMap A (A ⊗[R] I.ResidueField)) f) ↔
    ∀ i < Module.finrank R A, (LinearMap.charpoly ((Algebra.lmul R A) f)).coeff i ∈ I

Proof state:
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
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
⊢ IsNilpotent ((algebraMap A (A ⊗[R] I.ResidueField)) f) ↔
    ∀ i < Module.finrank R A, (LinearMap.charpoly ((Algebra.lmul R A) f)).coeff i ∈ I

Tactic:
rw [← IsNilpotent.map_iff (Algebra.TensorProduct.comm R A I.ResidueField).injective]

Result:
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
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
⊢ IsNilpotent ((Algebra.TensorProduct.comm R A I.ResidueField) ((algebraMap A (A ⊗[R] I.ResidueField)) f)) ↔
    ∀ i < Module.finrank R A, (LinearMap.charpoly ((Algebra.lmul R A) f)).coeff i ∈ I

Proof state:
case inr.mpr.a
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
⊢ (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:
obtain (hi | hi) := eq_or_ne i ((LinearMap.mul R A) f).charpoly.natDegree

Result:
case inr.mpr.a.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

case inr.mpr.a.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

Proof state:
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
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
⊢ IsNilpotent ((Algebra.TensorProduct.comm R A I.ResidueField) ((algebraMap A (A ⊗[R] I.ResidueField)) f)) ↔
    ∀ i < Module.finrank R A, (LinearMap.charpoly ((Algebra.lmul R A) f)).coeff i ∈ I

Tactic:
simp only [Algebra.TensorProduct.algebraMap_apply, Algebra.id.map_eq_id, RingHom.id_apply,
  Algebra.coe_lmul_eq_mul, Algebra.TensorProduct.comm_tmul]

Result:
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
this : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A
⊢ IsNilpotent (1 ⊗ₜ[R] f) ↔ ∀ i < Module.finrank R A, ((LinearMap.mul R A) f).charpoly.coeff i ∈ I

Proof state:
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
⊢ Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A

Tactic:
rw [Module.finrank_tensorProduct, Module.finrank_self, one_mul]

Result:
no goals