Ramification index and inertia degree

Given P : Ideal S lying over p : Ideal R for the ring extension f : R →+* S (assuming P and p are prime or maximal where needed), the ramification index Ideal.ramificationIdx f p P is the multiplicity of P in map f p, and the inertia degree Ideal.inertiaDeg f p P is the degree of the field extension (S / P) : (R / p).

Main results

The main theorem Ideal.sum_ramification_inertia states that for all coprime P lying over p, Σ P, ramification_idx f p P * inertia_deg f p P equals the degree of the field extension Frac(S) : Frac(R).

Implementation notes

Often the above theory is set up in the case where:

• R is the ring of integers of a number field K,
• L is a finite separable extension of K,
• S is the integral closure of R in L,
• p and P are maximal ideals,
• P is an ideal lying over p

We will try to relax the above hypotheses as much as possible.

Notation

In this file, e stands for the ramification index and f for the inertia degree of P over p, leaving p and P implicit.

theorem Ideal.FinrankQuotientMap.span_eq_top {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) {K : Type u_1} [Field K] [Algebra R K] {L : Type u_2} [Field L] [Algebra S L] [IsFractionRing S L] [IsDomain R] [IsDomain S] [Algebra K L] [Module.Finite R S] [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Algebra.IsAlgebraic R S] [Module.IsTorsionFree R K] (hp : p ≠ ⊤) (b : Set S) (hb' : Submodule.span R b ⊔ Submodule.restrictScalars R (map (algebraMap R S) p) = ⊤) : Submodule.span K (⇑(algebraMap S L) '' b) = ⊤

If b mod p spans S/p as R/p-space, then b itself spans Frac(S) as K-space.

Here,

• p is an ideal of R such that R / p is nontrivial
• K is a field that has an embedding of R (in particular we can take K = Frac(R))
• L is a field extension of K
• S is the integral closure of R in L

More precisely, we avoid quotients in this statement and instead require that b ∪ pS spans S.

theorem Ideal.FinrankQuotientMap.linearIndependent_of_nontrivial {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (K : Type u_1) [Field K] [Algebra R K] {V : Type u_3} {V' : Type u_4} {V'' : Type u_5} [AddCommGroup V] [Module R V] [Module K V] [IsScalarTower R K V] [AddCommGroup V'] [Module R V'] [Module S V'] [IsScalarTower R S V'] [AddCommGroup V''] [Module R V''] [hRK : IsFractionRing R K] [IsDedekindDomain R] (hRS : RingHom.ker (algebraMap R S) ≠ ⊤) (F : V'' →ₗ[R] V) (hf : Function.Injective ⇑F) (f' : V'' →ₗ[R] V') {ι : Type u_6} {b : ι → V''} (hb' : LinearIndependent S (⇑f' ∘ b)) : LinearIndependent K (⇑F ∘ b)

Let V be a vector space over K = Frac(R), S / R a ring extension and V' a module over S. If b, in the intersection V'' of V and V', is linear independent over S in V', then it is linear independent over R in V.

The statement we prove is actually slightly more general:

• it suffices that the inclusion algebraMap R S : R → S is nontrivial
• the function f' : V'' → V' doesn't need to be injective

theorem Ideal.finrank_quotient_map {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (K : Type u_1) [Field K] [Algebra R K] (L : Type u_2) [Field L] [Algebra S L] [IsFractionRing S L] [hRK : IsFractionRing R K] [IsDomain S] [IsDedekindDomain R] [Algebra K L] [Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L] [hp : p.IsMaximal] [Module.Finite R S] : Module.finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = Module.finrank K L

If p is a maximal ideal of R, and S is the integral closure of R in L, then the dimension [S/pS : R/p] is equal to [Frac(S) : Frac(R)].

@[implicit_reducible]

noncomputable instance Ideal.Quotient.algebraQuotientPowRamificationIdx {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) : Algebra (R ⧸ p) (S ⧸ P ^ p.ramificationIdx P)

R / p has a canonical map to S / (P ^ e), where e is the ramification index of P over p.

Equations

• Ideal.Quotient.algebraQuotientPowRamificationIdx p P = Ideal.Quotient.algebraQuotientOfLEComap ⋯

@[simp]

theorem Ideal.Quotient.algebraMap_quotient_pow_ramificationIdx {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) (x : R) : (algebraMap (R ⧸ p) (S ⧸ P ^ p.ramificationIdx P)) ((mk p) x) = (mk (P ^ p.ramificationIdx P)) ((algebraMap R S) x)

@[implicit_reducible]

def Ideal.Quotient.algebraQuotientOfRamificationIdxNeZero {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [hfp : NeZero (p.ramificationIdx P)] : Algebra (R ⧸ p) (S ⧸ P)

If P lies over p, then R / p has a canonical map to S / P.

This can't be an instance since the map f : R → S is generally not inferable.

Equations

• Ideal.Quotient.algebraQuotientOfRamificationIdxNeZero p P = Ideal.Quotient.algebraQuotientOfLEComap ⋯

@[simp]

theorem Ideal.Quotient.algebraMap_quotient_of_ramificationIdx_neZero {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [NeZero (p.ramificationIdx P)] (x : R) : (algebraMap (R ⧸ p) (S ⧸ P)) ((mk p) x) = (mk P) ((algebraMap R S) x)

noncomputable def Ideal.powQuotSuccInclusion {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) (i : ℕ) : ↥(map (Quotient.mk (P ^ p.ramificationIdx P)) (P ^ (i + 1))) →ₗ[R ⧸ p] ↥(map (Quotient.mk (P ^ p.ramificationIdx P)) (P ^ i))

The inclusion (P^(i + 1) / P^e) ⊂ (P^i / P^e).

Equations

• p.powQuotSuccInclusion P i = { toFun := fun (x : ↥(Ideal.map (Ideal.Quotient.mk (P ^ p.ramificationIdx P)) (P ^ (i + 1)))) => ⟨↑x, ⋯⟩, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem Ideal.powQuotSuccInclusion_apply_coe {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) (i : ℕ) (x : ↥(map (Quotient.mk (P ^ p.ramificationIdx P)) (P ^ (i + 1)))) : ↑((p.powQuotSuccInclusion P i) x) = ↑x

theorem Ideal.powQuotSuccInclusion_injective {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) (i : ℕ) : Function.Injective ⇑(p.powQuotSuccInclusion P i)

noncomputable def Ideal.quotientToQuotientRangePowQuotSuccAux {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) : S ⧸ P → ↥(map (Quotient.mk (P ^ p.ramificationIdx P)) (P ^ i)) ⧸ (p.powQuotSuccInclusion P i).range

S ⧸ P embeds into the quotient by P^(i+1) ⧸ P^e as a subspace of P^i ⧸ P^e. See quotientToQuotientRangePowQuotSucc for this as a linear map, and quotientRangePowQuotSuccInclusionEquiv for this as a linear equivalence.

Equations

• p.quotientToQuotientRangePowQuotSuccAux P a_mem = Quotient.map' (fun (x : S) => ⟨(Ideal.Quotient.mk (P ^ p.ramificationIdx P)) (a * x), ⋯⟩) ⋯

theorem Ideal.quotientToQuotientRangePowQuotSuccAux_mk {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S) : p.quotientToQuotientRangePowQuotSuccAux P a_mem (Submodule.Quotient.mk x) = Submodule.Quotient.mk ⟨(Quotient.mk (P ^ p.ramificationIdx P)) (a * x), ⋯⟩

noncomputable def Ideal.quotientToQuotientRangePowQuotSucc {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [hfp : NeZero (p.ramificationIdx P)] {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) : S ⧸ P →ₗ[R ⧸ p] ↥(map (Quotient.mk (P ^ p.ramificationIdx P)) (P ^ i)) ⧸ (p.powQuotSuccInclusion P i).range

S ⧸ P embeds into the quotient by P^(i+1) ⧸ P^e as a subspace of P^i ⧸ P^e.

Equations

• p.quotientToQuotientRangePowQuotSucc P a_mem = { toFun := p.quotientToQuotientRangePowQuotSuccAux P a_mem, map_add' := ⋯, map_smul' := ⋯ }

theorem Ideal.quotientToQuotientRangePowQuotSucc_mk {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [hfp : NeZero (p.ramificationIdx P)] {i : ℕ} {a : S} (a_mem : a ∈ P ^ i) (x : S) : (p.quotientToQuotientRangePowQuotSucc P a_mem) (Submodule.Quotient.mk x) = Submodule.Quotient.mk ⟨(Quotient.mk (P ^ p.ramificationIdx P)) (a * x), ⋯⟩

theorem Ideal.quotientToQuotientRangePowQuotSucc_injective {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [hfp : NeZero (p.ramificationIdx P)] [IsDedekindDomain S] [P.IsPrime] {i : ℕ} (hi : i < p.ramificationIdx P) {a : S} (a_mem : a ∈ P ^ i) (a_notMem : a ∉ P ^ (i + 1)) : Function.Injective ⇑(p.quotientToQuotientRangePowQuotSucc P a_mem)

theorem Ideal.quotientToQuotientRangePowQuotSucc_surjective {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [hfp : NeZero (p.ramificationIdx P)] [IsDedekindDomain S] (hP0 : P ≠ ⊥) [hP : P.IsPrime] {i : ℕ} (hi : i < p.ramificationIdx P) {a : S} (a_mem : a ∈ P ^ i) (a_notMem : a ∉ P ^ (i + 1)) : Function.Surjective ⇑(p.quotientToQuotientRangePowQuotSucc P a_mem)

noncomputable def Ideal.quotientRangePowQuotSuccInclusionEquiv {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [hfp : NeZero (p.ramificationIdx P)] [IsDedekindDomain S] [P.IsPrime] (hP : P ≠ ⊥) {i : ℕ} (hi : i < p.ramificationIdx P) : (↥(map (Quotient.mk (P ^ p.ramificationIdx P)) (P ^ i)) ⧸ (p.powQuotSuccInclusion P i).range) ≃ₗ[R ⧸ p] S ⧸ P

Quotienting P^i / P^e by its subspace P^(i+1) ⧸ P^e is R ⧸ p-linearly isomorphic to S ⧸ P.

Equations

• One or more equations did not get rendered due to their size.

theorem Ideal.rank_pow_quot_aux {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [hfp : NeZero (p.ramificationIdx P)] [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥) {i : ℕ} (hi : i < p.ramificationIdx P) : Module.rank (R ⧸ p) ↥(map (Quotient.mk (P ^ p.ramificationIdx P)) (P ^ i)) = Module.rank (R ⧸ p) (S ⧸ P) + Module.rank (R ⧸ p) ↥(map (Quotient.mk (P ^ p.ramificationIdx P)) (P ^ (i + 1)))

Since the inclusion (P^(i + 1) / P^e) ⊂ (P^i / P^e) has a kernel isomorphic to P / S, [P^i / P^e : R / p] = [P^(i+1) / P^e : R / p] + [P / S : R / p]

theorem Ideal.rank_pow_quot {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [hfp : NeZero (p.ramificationIdx P)] [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥) (i : ℕ) (hi : i ≤ p.ramificationIdx P) : Module.rank (R ⧸ p) ↥(map (Quotient.mk (P ^ p.ramificationIdx P)) (P ^ i)) = (p.ramificationIdx P - i) • Module.rank (R ⧸ p) (S ⧸ P)

theorem Ideal.rank_prime_pow_ramificationIdx {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥) (he : p.ramificationIdx P ≠ 0) : Module.rank (R ⧸ p) (S ⧸ P ^ p.ramificationIdx P) = p.ramificationIdx P • Module.rank (R ⧸ p) (S ⧸ P)

If p is a maximal ideal of R, S extends R and P^e lies over p, then the dimension [S/(P^e) : R/p] is equal to e * [S/P : R/p].

theorem Ideal.finrank_prime_pow_ramificationIdx {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) (P : Ideal S) [IsDedekindDomain S] (hP0 : P ≠ ⊥) [p.IsMaximal] [P.IsPrime] (he : p.ramificationIdx P ≠ 0) : Module.finrank (R ⧸ p) (S ⧸ P ^ p.ramificationIdx P) = p.ramificationIdx P * Module.finrank (R ⧸ p) (S ⧸ P)

If p is a maximal ideal of R, S extends R and P^e lies over p, then the dimension [S/(P^e) : R/p], as a natural number, is equal to e * [S/P : R/p].

Properties of the factors of p.map (algebraMap R S)

theorem Ideal.Factors.ne_bot {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) [IsDedekindDomain S] (P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) : ↑P ≠ ⊥

instance Ideal.Factors.isPrime {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) [IsDedekindDomain S] (P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) : (↑P).IsPrime

theorem Ideal.Factors.ramificationIdx_ne_zero {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) [IsDedekindDomain S] (P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) : p.ramificationIdx ↑P ≠ 0

instance Ideal.Factors.fact_ramificationIdx_neZero {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) [IsDedekindDomain S] (P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) : NeZero (p.ramificationIdx ↑P)

instance Ideal.Factors.isScalarTower {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) [IsDedekindDomain S] (P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) : IsScalarTower R (R ⧸ p) (S ⧸ ↑P)

instance Ideal.Factors.liesOver {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) [IsDedekindDomain S] [p.IsMaximal] (P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) : (↑P).LiesOver p

theorem Ideal.Factors.finrank_pow_ramificationIdx {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) [IsDedekindDomain S] [p.IsMaximal] (P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) : Module.finrank (R ⧸ p) (S ⧸ ↑P ^ p.ramificationIdx ↑P) = p.ramificationIdx ↑P * p.inertiaDeg ↑P

instance Ideal.Factors.finiteDimensional_quotient_pow {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] (p : Ideal R) [IsDedekindDomain S] [Module.Finite R S] [p.IsMaximal] (P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) : FiniteDimensional (R ⧸ p) (S ⧸ ↑P ^ p.ramificationIdx ↑P)

noncomputable def Ideal.Factors.piQuotientEquiv {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] [IsDedekindDomain S] (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) : S ⧸ map (algebraMap R S) p ≃+* ((P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) → S ⧸ ↑P ^ p.ramificationIdx ↑P)

Chinese remainder theorem for a ring of integers: if the prime ideal p : Ideal R factors in S as ∏ i, P i ^ e i, then S ⧸ I factors as Π i, R ⧸ (P i ^ e i).

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Ideal.Factors.piQuotientEquiv_mk {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] [IsDedekindDomain S] (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) (x : S) : (piQuotientEquiv p hp) ((Quotient.mk (map (algebraMap R S) p)) x) = fun (x_1 : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) => (Quotient.mk (↑x_1 ^ p.ramificationIdx ↑x_1)) x

@[simp]

theorem Ideal.Factors.piQuotientEquiv_map {R : Type u} [CommRing R] {S : Type v} [CommRing S] [Algebra R S] [IsDedekindDomain S] (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) (x : R) : (piQuotientEquiv p hp) ((algebraMap R (S ⧸ map (algebraMap R S) p)) x) = fun (x_1 : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) => (Quotient.mk (↑x_1 ^ p.ramificationIdx ↑x_1)) ((algebraMap R S) x)

noncomputable def Ideal.Factors.piQuotientLinearEquiv {R : Type u} [CommRing R] (S : Type v) [CommRing S] [Algebra R S] [IsDedekindDomain S] (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) : (S ⧸ map (algebraMap R S) p) ≃ₗ[R ⧸ p] (P : ↥(UniqueFactorizationMonoid.factors (map (algebraMap R S) p)).toFinset) → S ⧸ ↑P ^ p.ramificationIdx ↑P

Chinese remainder theorem for a ring of integers: if the prime ideal p : Ideal R factors in S as ∏ i, P i ^ e i, then S ⧸ I factors R ⧸ I-linearly as Π i, R ⧸ (P i ^ e i).

Equations

• One or more equations did not get rendered due to their size.

theorem Ideal.sum_ramification_inertia {R : Type u} [CommRing R] (S : Type v) [CommRing S] [Algebra R S] [IsDedekindDomain S] (K : Type u_1) (L : Type u_2) [Field K] [Field L] [IsDedekindDomain R] [Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L] [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Module.Finite R S] {p : Ideal R} [p.IsMaximal] (hp0 : p ≠ ⊥) : ∑ P ∈ primesOverFinset p S, p.ramificationIdx P * p.inertiaDeg P = Module.finrank K L

The fundamental identity of ramification index e and inertia degree f: for P ranging over the primes lying over p, ∑ P, e P * f P = [Frac(S) : Frac(R)]; here S is a finite R-module (and thus Frac(S) : Frac(R) is a finite extension) and p is maximal.

theorem Ideal.inertiaDeg_le_finrank {R : Type u} [CommRing R] (S : Type v) [CommRing S] [Algebra R S] [IsDedekindDomain S] (K : Type u_1) (L : Type u_2) [Field K] [Field L] [IsDedekindDomain R] [Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L] [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Module.Finite R S] [NoZeroSMulDivisors R S] {p : Ideal R} [p.IsMaximal] (P : Ideal S) [hP₁ : P.IsPrime] [hP₂ : P.LiesOver p] (hp0 : p ≠ ⊥) : p.inertiaDeg P ≤ Module.finrank K L

theorem Ideal.ramificationIdx_le_finrank {R : Type u} [CommRing R] (S : Type v) [CommRing S] [Algebra R S] [IsDedekindDomain S] (K : Type u_1) (L : Type u_2) [Field K] [Field L] [IsDedekindDomain R] [Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L] [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Module.Finite R S] [NoZeroSMulDivisors R S] {p : Ideal R} [p.IsMaximal] (P : Ideal S) [hP₁ : P.IsPrime] [hP₂ : P.LiesOver p] : p.ramificationIdx P ≤ Module.finrank K L

theorem Ideal.card_primesOverFinset_le_finrank {R : Type u} [CommRing R] (S : Type v) [CommRing S] [Algebra R S] [IsDedekindDomain S] (K : Type u_1) (L : Type u_2) [Field K] [Field L] [IsDedekindDomain R] [Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L] [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Module.Finite R S] [NoZeroSMulDivisors R S] {p : Ideal R} [p.IsMaximal] (hp0 : p ≠ ⊥) : (primesOverFinset p S).card ≤ Module.finrank K L

theorem Ideal.ramificationIdx_mul_inertiaDeg_of_isLocalRing {R : Type u} [CommRing R] (S : Type v) [CommRing S] [Algebra R S] [IsDedekindDomain S] (K : Type u_1) (L : Type u_2) [Field K] [Field L] [IsDedekindDomain R] [Algebra R K] [IsFractionRing R K] [Algebra S L] [IsFractionRing S L] [Algebra K L] [Algebra R L] [IsScalarTower R S L] [IsScalarTower R K L] [Module.Finite R S] [IsLocalRing S] {p : Ideal R} [p.IsMaximal] (hp0 : p ≠ ⊥) : p.ramificationIdx (IsLocalRing.maximalIdeal S) * p.inertiaDeg (IsLocalRing.maximalIdeal S) = Module.finrank K L

Ideal.sum_ramification_inertia, in the local (DVR) case.