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/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import group_theory.quotient_group
import ring_theory.dedekind_domain.ideal
/-!
# The ideal class group
This file defines the ideal class group `class_group R K` of fractional ideals of `R`
inside `A`'s field of fractions `K`.
## Main definitions
- `to_principal_ideal` sends an invertible `x : K` to an invertible fractional ideal
- `class_group` is the quotient of invertible fractional ideals modulo `to_principal_ideal.range`
- `class_group.mk0` sends a nonzero integral ideal in a Dedekind domain to its class
## Main results
- `class_group.mk0_eq_mk0_iff` shows the equivalence with the "classical" definition,
where `I ~ J` iff `x I = y J` for `x y ≠ (0 : R)`
-/
variables {R K L : Type*} [comm_ring R]
variables [field K] [field L] [decidable_eq L]
variables [algebra R K] [is_fraction_ring R K]
variables [algebra K L] [finite_dimensional K L]
variables [algebra R L] [is_scalar_tower R K L]
open_locale non_zero_divisors
open is_localization is_fraction_ring fractional_ideal units
section
variables (R K)
/-- `to_principal_ideal R K x` sends `x ≠ 0 : K` to the fractional `R`-ideal generated by `x` -/
@[irreducible]
def to_principal_ideal : Kˣ →* (fractional_ideal R⁰ K)ˣ :=
{ to_fun := λ x,
⟨span_singleton _ x,
span_singleton _ x⁻¹,
by simp only [span_singleton_one, units.mul_inv', span_singleton_mul_span_singleton],
by simp only [span_singleton_one, units.inv_mul', span_singleton_mul_span_singleton]⟩,
map_mul' := λ x y, ext
(by simp only [units.coe_mk, units.coe_mul, span_singleton_mul_span_singleton]),
map_one' := ext (by simp only [span_singleton_one, units.coe_mk, units.coe_one]) }
local attribute [semireducible] to_principal_ideal
variables {R K}
@[simp] lemma coe_to_principal_ideal (x : Kˣ) :
(to_principal_ideal R K x : fractional_ideal R⁰ K) = span_singleton _ x :=
rfl
@[simp] lemma to_principal_ideal_eq_iff {I : (fractional_ideal R⁰ K)ˣ} {x : Kˣ} :
to_principal_ideal R K x = I ↔ span_singleton R⁰ (x : K) = I :=
units.ext_iff
end
instance principal_ideals.normal : (to_principal_ideal R K).range.normal :=
subgroup.normal_of_comm _
section
variables (R K)
/-- The ideal class group of `R` in a field of fractions `K`
is the group of invertible fractional ideals modulo the principal ideals. -/
@[derive(comm_group)]
def class_group := (fractional_ideal R⁰ K)ˣ ⧸ (to_principal_ideal R K).range
instance : inhabited (class_group R K) := ⟨1⟩
variables {R} [is_domain R]
/-- Send a nonzero integral ideal to an invertible fractional ideal. -/
@[simps]
noncomputable def fractional_ideal.mk0 [is_dedekind_domain R] :
(ideal R)⁰ →* (fractional_ideal R⁰ K)ˣ :=
{ to_fun := λ I, units.mk0 I ((fractional_ideal.coe_to_fractional_ideal_ne_zero (le_refl R⁰)).mpr
(mem_non_zero_divisors_iff_ne_zero.mp I.2)),
map_one' := by simp,
map_mul' := λ x y, by simp }
/-- Send a nonzero ideal to the corresponding class in the class group. -/
@[simps]
noncomputable def class_group.mk0 [is_dedekind_domain R] :
(ideal R)⁰ →* class_group R K :=
(quotient_group.mk' _).comp (fractional_ideal.mk0 K)
variables {K}
lemma class_group.mk0_eq_mk0_iff_exists_fraction_ring [is_dedekind_domain R] {I J : (ideal R)⁰} :
class_group.mk0 K I = class_group.mk0 K J ↔
∃ (x ≠ (0 : K)), span_singleton R⁰ x * I = J :=
begin
simp only [class_group.mk0, monoid_hom.comp_apply, quotient_group.mk'_eq_mk'],
split,
{ rintros ⟨_, ⟨x, rfl⟩, hx⟩,
refine ⟨x, x.ne_zero, _⟩,
simpa only [mul_comm, coe_mk0, monoid_hom.to_fun_eq_coe, coe_to_principal_ideal, units.coe_mul]
using congr_arg (coe : _ → fractional_ideal R⁰ K) hx },
{ rintros ⟨x, hx, eq_J⟩,
refine ⟨_, ⟨units.mk0 x hx, rfl⟩, units.ext _⟩,
simpa only [fractional_ideal.mk0_apply, units.coe_mk0, mul_comm, coe_to_principal_ideal,
coe_coe, units.coe_mul] using eq_J }
end
lemma class_group.mk0_eq_mk0_iff [is_dedekind_domain R] {I J : (ideal R)⁰} :
class_group.mk0 K I = class_group.mk0 K J ↔
∃ (x y : R) (hx : x ≠ 0) (hy : y ≠ 0), ideal.span {x} * (I : ideal R) = ideal.span {y} * J :=
begin
refine class_group.mk0_eq_mk0_iff_exists_fraction_ring.trans ⟨_, _⟩,
{ rintros ⟨z, hz, h⟩,
obtain ⟨x, ⟨y, hy⟩, rfl⟩ := is_localization.mk'_surjective R⁰ z,
refine ⟨x, y, _, mem_non_zero_divisors_iff_ne_zero.mp hy, _⟩,
{ rintro hx, apply hz,
rw [hx, is_fraction_ring.mk'_eq_div, (algebra_map R K).map_zero, zero_div] },
{ exact (fractional_ideal.mk'_mul_coe_ideal_eq_coe_ideal K hy).mp h } },
{ rintros ⟨x, y, hx, hy, h⟩,
have hy' : y ∈ R⁰ := mem_non_zero_divisors_iff_ne_zero.mpr hy,
refine ⟨is_localization.mk' K x ⟨y, hy'⟩, _, _⟩,
{ contrapose! hx,
rwa [is_localization.mk'_eq_iff_eq_mul, zero_mul, ← (algebra_map R K).map_zero,
(is_fraction_ring.injective R K).eq_iff] at hx },
{ exact (fractional_ideal.mk'_mul_coe_ideal_eq_coe_ideal K hy').mpr h } },
end
lemma class_group.mk0_surjective [is_dedekind_domain R] :
function.surjective (class_group.mk0 K : (ideal R)⁰ → class_group R K) :=
begin
rintros ⟨I⟩,
obtain ⟨a, a_ne_zero', ha⟩ := I.1.2,
have a_ne_zero := mem_non_zero_divisors_iff_ne_zero.mp a_ne_zero',
have fa_ne_zero : (algebra_map R K) a ≠ 0 :=
is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors a_ne_zero',
refine ⟨⟨{ carrier := { x | (algebra_map R K a)⁻¹ * algebra_map R K x ∈ I.1 }, .. }, _⟩, _⟩,
{ simp only [ring_hom.map_add, set.mem_set_of_eq, mul_zero, ring_hom.map_mul, mul_add],
exact λ _ _ ha hb, submodule.add_mem I ha hb },
{ simp only [ring_hom.map_zero, set.mem_set_of_eq, mul_zero, ring_hom.map_mul],
exact submodule.zero_mem I },
{ intros c _ hb,
simp only [smul_eq_mul, set.mem_set_of_eq, mul_zero, ring_hom.map_mul, mul_add,
mul_left_comm ((algebra_map R K) a)⁻¹],
rw ← algebra.smul_def c,
exact submodule.smul_mem I c hb },
{ rw [mem_non_zero_divisors_iff_ne_zero, submodule.zero_eq_bot, submodule.ne_bot_iff],
obtain ⟨x, x_ne, x_mem⟩ := exists_ne_zero_mem_is_integer I.ne_zero,
refine ⟨a * x, _, mul_ne_zero a_ne_zero x_ne⟩,
change ((algebra_map R K) a)⁻¹ * (algebra_map R K) (a * x) ∈ I.1,
rwa [ring_hom.map_mul, ← mul_assoc, inv_mul_cancel fa_ne_zero, one_mul] },
{ symmetry,
apply quotient.sound,
change setoid.r _ _,
rw quotient_group.left_rel_apply,
refine ⟨units.mk0 (algebra_map R K a) fa_ne_zero, _⟩,
apply @mul_left_cancel _ _ I,
rw [← mul_assoc, mul_right_inv, one_mul, eq_comm, mul_comm I],
apply units.ext,
simp only [monoid_hom.coe_mk, subtype.coe_mk, ring_hom.map_mul, coe_coe,
units.coe_mul, coe_to_principal_ideal, coe_mk0,
fractional_ideal.eq_span_singleton_mul],
split,
{ intros zJ' hzJ',
obtain ⟨zJ, hzJ : (algebra_map R K a)⁻¹ * algebra_map R K zJ ∈ ↑I, rfl⟩ :=
(mem_coe_ideal R⁰).mp hzJ',
refine ⟨_, hzJ, _⟩,
rw [← mul_assoc, mul_inv_cancel fa_ne_zero, one_mul] },
{ intros zI' hzI',
obtain ⟨y, hy⟩ := ha zI' hzI',
rw [← algebra.smul_def, fractional_ideal.mk0_apply, coe_mk0, coe_coe, mem_coe_ideal],
refine ⟨y, _, hy⟩,
show (algebra_map R K a)⁻¹ * algebra_map R K y ∈ (I : fractional_ideal R⁰ K),
rwa [hy, algebra.smul_def, ← mul_assoc, inv_mul_cancel fa_ne_zero, one_mul] } }
end
end
lemma class_group.mk_eq_one_iff
{I : (fractional_ideal R⁰ K)ˣ} :
quotient_group.mk' (to_principal_ideal R K).range I = 1 ↔
(I : submodule R K).is_principal :=
begin
rw [← (quotient_group.mk' _).map_one, eq_comm, quotient_group.mk'_eq_mk'],
simp only [exists_prop, one_mul, exists_eq_right, to_principal_ideal_eq_iff,
monoid_hom.mem_range, coe_coe],
refine ⟨λ ⟨x, hx⟩, ⟨⟨x, by rw [← hx, coe_span_singleton]⟩⟩, _⟩,
unfreezingI { intros hI },
obtain ⟨x, hx⟩ := @submodule.is_principal.principal _ _ _ _ _ _ hI,
have hx' : (I : fractional_ideal R⁰ K) = span_singleton R⁰ x,
{ apply subtype.coe_injective, rw [hx, coe_span_singleton] },
refine ⟨units.mk0 x _, _⟩,
{ intro x_eq, apply units.ne_zero I, simp [hx', x_eq] },
simp [hx']
end
variables [is_domain R]
lemma class_group.mk0_eq_one_iff [is_dedekind_domain R]
{I : ideal R} (hI : I ∈ (ideal R)⁰) :
class_group.mk0 K ⟨I, hI⟩ = 1 ↔ I.is_principal :=
class_group.mk_eq_one_iff.trans (coe_submodule_is_principal R K)
/-- The class group of principal ideal domain is finite (in fact a singleton).
TODO: generalize to Dedekind domains -/
instance [is_principal_ideal_ring R] :
fintype (class_group R K) :=
{ elems := {1},
complete :=
begin
rintros ⟨I⟩,
rw [finset.mem_singleton],
exact class_group.mk_eq_one_iff.mpr (I : fractional_ideal R⁰ K).is_principal
end }
/-- The class number of a principal ideal domain is `1`. -/
lemma card_class_group_eq_one [is_principal_ideal_ring R] :
fintype.card (class_group R K) = 1 :=
begin
rw fintype.card_eq_one_iff,
use 1,
rintros ⟨I⟩,
exact class_group.mk_eq_one_iff.mpr (I : fractional_ideal R⁰ K).is_principal
end
/-- The class number is `1` iff the ring of integers is a principal ideal domain. -/
lemma card_class_group_eq_one_iff [is_dedekind_domain R] [fintype (class_group R K)] :
fintype.card (class_group R K) = 1 ↔ is_principal_ideal_ring R :=
begin
split, swap, { introsI, convert card_class_group_eq_one, assumption, assumption, },
rw fintype.card_eq_one_iff,
rintros ⟨I, hI⟩,
have eq_one : ∀ J : class_group R K, J = 1 := λ J, trans (hI J) (hI 1).symm,
refine ⟨λ I, _⟩,
by_cases hI : I = ⊥,
{ rw hI, exact bot_is_principal },
exact (class_group.mk0_eq_one_iff (mem_non_zero_divisors_iff_ne_zero.mpr hI)).mp (eq_one _),
end
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