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number_theory.ramification_inertia

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.ramification_idx f p P is the multiplicity of P in map f p, and the inertia degree ideal.inertia_deg f p P is the degree of the field extension (S / P) : (R / p).

TODO (#12287) #

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:

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.

noncomputable def ideal.ramification_idx {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) :

The ramification index of P over p is the largest exponent n such that p is contained in P^n.

In particular, if p is not contained in P^n, then the ramification index is 0.

If there is no largest such n (e.g. because p = ⊥), then ramification_idx is defined to be 0.

Equations
theorem ideal.ramification_idx_eq_find {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} (h : ∃ (n : ), ∀ (k : ), ideal.map f p P ^ kk n) :
theorem ideal.ramification_idx_eq_zero {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} (h : ∀ (n : ), ∃ (k : ), ideal.map f p P ^ k n < k) :
theorem ideal.ramification_idx_spec {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} {n : } (hle : ideal.map f p P ^ n) (hgt : ¬ideal.map f p P ^ (n + 1)) :
theorem ideal.ramification_idx_lt {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} {n : } (hgt : ¬ideal.map f p P ^ n) :
@[simp]
theorem ideal.ramification_idx_bot {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {P : ideal S} :
@[simp]
theorem ideal.ramification_idx_of_not_le {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} (h : ¬ideal.map f p P) :
theorem ideal.ramification_idx_ne_zero {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} {e : } (he : e 0) (hle : ideal.map f p P ^ e) (hnle : ¬ideal.map f p P ^ (e + 1)) :
theorem ideal.le_pow_of_le_ramification_idx {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} {n : } (hn : n ideal.ramification_idx f p P) :
ideal.map f p P ^ n
theorem ideal.le_pow_ramification_idx {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} :
theorem ideal.le_comap_of_ramification_idx_ne_zero {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} (h : ideal.ramification_idx f p P 0) :
theorem ideal.is_dedekind_domain.ramification_idx_ne_zero {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] {f : R →+* S} {p : ideal R} {P : ideal S} [is_domain S] [is_dedekind_domain S] (hp0 : ideal.map f p ) (hP : P.is_prime) (le : ideal.map f p P) :
noncomputable def ideal.inertia_deg {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hp : p.is_maximal] :

The inertia degree of P : ideal S lying over p : ideal R is the degree of the extension (S / P) : (R / p).

We do not assume P lies over p in the definition; we return 0 instead.

See inertia_deg_algebra_map for the common case where f = algebra_map R S and there is an algebra structure R / p → S / P.

Equations
@[simp]
theorem ideal.inertia_deg_of_subsingleton {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hp : p.is_maximal] [hQ : subsingleton (S P)] :
@[simp]
theorem ideal.inertia_deg_algebra_map {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (p : ideal R) (P : ideal S) [algebra R S] [algebra (R p) (S P)] [is_scalar_tower R (R p) (S P)] [hp : p.is_maximal] :
theorem ideal.finrank_quotient_map.linear_independent_of_nontrivial {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] [algebra R S] (K : Type u_1) [field K] [algebra R K] [hRK : is_fraction_ring R K] {V : Type u_3} {V' : Type u_4} {V'' : Type u_5} [add_comm_group V] [module R V] [module K V] [is_scalar_tower R K V] [add_comm_group V'] [module R V'] [module S V'] [is_scalar_tower R S V'] [add_comm_group V''] [module R V''] [is_domain R] [is_dedekind_domain R] (hRS : ring_hom.ker (algebra_map R S) ) (f : V'' →ₗ[R] V) (hf : function.injective f) (f' : V'' →ₗ[R] V') {ι : Type u_2} {b : ι → V''} (hb' : linear_independent S (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 algebra_map R S : R → S is nontrivial
  • the function f' : V'' → V' doesn't need to be injective
theorem ideal.finrank_quotient_map.span_eq_top {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (p : ideal R) [algebra R S] {K : Type u_1} [field K] [algebra R K] {L : Type u_2} [field L] [algebra S L] [is_fraction_ring S L] [is_domain R] [is_domain S] [algebra K L] [is_noetherian R S] [algebra R L] [is_scalar_tower R S L] [is_scalar_tower R K L] [is_integral_closure S R L] [no_zero_smul_divisors R K] (hp : p ) (b : set S) (hb' : submodule.span R b submodule.restrict_scalars R (ideal.map (algebra_map R S) p) = ) :

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.finrank_quotient_map {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (p : ideal R) [algebra R S] (K : Type u_1) [field K] [algebra R K] [hRK : is_fraction_ring R K] (L : Type u_2) [field L] [algebra S L] [is_fraction_ring S L] [is_domain R] [is_domain S] [is_dedekind_domain R] [algebra K L] [algebra R L] [is_scalar_tower R K L] [is_scalar_tower R S L] [is_integral_closure S R L] [hp : p.is_maximal] [is_noetherian R S] :

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)].

@[protected, instance]
noncomputable def ideal.quotient.algebra_quotient_pow_ramification_idx {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) :

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

Equations
@[simp]
theorem ideal.quotient.algebra_map_quotient_pow_ramification_idx {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) (x : R) :
def ideal.quotient.algebra_quotient_of_ramification_idx_ne_zero {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hfp : fact (ideal.ramification_idx f p P 0)] :
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 inferrable.

Equations
@[simp]
theorem ideal.quotient.algebra_map_quotient_of_ramification_idx_ne_zero {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hfp : fact (ideal.ramification_idx f p P 0)] (x : R) :
def ideal.pow_quot_succ_inclusion {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) (i : ) :

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

Equations
@[simp]
theorem ideal.pow_quot_succ_inclusion_apply_coe {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) (i : ) (x : (ideal.map (ideal.quotient.mk (P ^ ideal.ramification_idx f p P)) (P ^ (i + 1)))) :
theorem ideal.pow_quot_succ_inclusion_injective {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) (i : ) :
noncomputable def ideal.quotient_to_quotient_range_pow_quot_succ_aux {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) {i : } {a : S} (a_mem : a P ^ i) :

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

Equations
theorem ideal.quotient_to_quotient_range_pow_quot_succ_aux_mk {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) {i : } {a : S} (a_mem : a P ^ i) (x : S) :
noncomputable def ideal.quotient_to_quotient_range_pow_quot_succ {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hfp : fact (ideal.ramification_idx f p P 0)] {i : } {a : S} (a_mem : a P ^ i) :

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

Equations
theorem ideal.quotient_to_quotient_range_pow_quot_succ_mk {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hfp : fact (ideal.ramification_idx f p P 0)] {i : } {a : S} (a_mem : a P ^ i) (x : S) :
theorem ideal.quotient_to_quotient_range_pow_quot_succ_injective {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hfp : fact (ideal.ramification_idx f p P 0)] [is_domain S] [is_dedekind_domain S] [P.is_prime] {i : } (hi : i < ideal.ramification_idx f p P) {a : S} (a_mem : a P ^ i) (a_not_mem : a P ^ (i + 1)) :
theorem ideal.quotient_to_quotient_range_pow_quot_succ_surjective {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hfp : fact (ideal.ramification_idx f p P 0)] [is_domain S] [is_dedekind_domain S] (hP0 : P ) [hP : P.is_prime] {i : } (hi : i < ideal.ramification_idx f p P) {a : S} (a_mem : a P ^ i) (a_not_mem : a P ^ (i + 1)) :
noncomputable def ideal.quotient_range_pow_quot_succ_inclusion_equiv {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hfp : fact (ideal.ramification_idx f p P 0)] [is_domain S] [is_dedekind_domain S] [P.is_prime] (hP : P ) {i : } (hi : i < ideal.ramification_idx f p P) :

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

Equations
theorem ideal.dim_pow_quot_aux {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hfp : fact (ideal.ramification_idx f p P 0)] [is_domain S] [is_dedekind_domain S] [p.is_maximal] [P.is_prime] (hP0 : P ) {i : } (hi : i < ideal.ramification_idx f p P) :

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.dim_pow_quot {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [hfp : fact (ideal.ramification_idx f p P 0)] [is_domain S] [is_dedekind_domain S] [p.is_maximal] [P.is_prime] (hP0 : P ) (i : ) (hi : i ideal.ramification_idx f p P) :
theorem ideal.dim_prime_pow_ramification_idx {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [is_domain S] [is_dedekind_domain S] [p.is_maximal] [P.is_prime] (hP0 : P ) (he : ideal.ramification_idx f p P 0) :

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_ramification_idx {R : Type u} [comm_ring R] {S : Type v} [comm_ring S] (f : R →+* S) (p : ideal R) (P : ideal S) [is_domain S] [is_dedekind_domain S] (hP0 : P ) [p.is_maximal] [P.is_prime] (he : ideal.ramification_idx f p P 0) :

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].