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fractional_ideal.lean
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fractional_ideal.lean
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/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import algebra.big_operators.finprod
import ring_theory.integral_closure
import ring_theory.localization.integer
import ring_theory.localization.submodule
import ring_theory.noetherian
import ring_theory.principal_ideal_domain
import tactic.field_simp
/-!
# Fractional ideals
This file defines fractional ideals of an integral domain and proves basic facts about them.
## Main definitions
Let `S` be a submonoid of an integral domain `R`, `P` the localization of `R` at `S`, and `f` the
natural ring hom from `R` to `P`.
* `is_fractional` defines which `R`-submodules of `P` are fractional ideals
* `fractional_ideal S P` is the type of fractional ideals in `P`
* `has_coe_t (ideal R) (fractional_ideal S P)` instance
* `comm_semiring (fractional_ideal S P)` instance:
the typical ideal operations generalized to fractional ideals
* `lattice (fractional_ideal S P)` instance
* `map` is the pushforward of a fractional ideal along an algebra morphism
Let `K` be the localization of `R` at `R⁰ = R \ {0}` (i.e. the field of fractions).
* `fractional_ideal R⁰ K` is the type of fractional ideals in the field of fractions
* `has_div (fractional_ideal R⁰ K)` instance:
the ideal quotient `I / J` (typically written $I : J$, but a `:` operator cannot be defined)
## Main statements
* `mul_left_mono` and `mul_right_mono` state that ideal multiplication is monotone
* `prod_one_self_div_eq` states that `1 / I` is the inverse of `I` if one exists
* `is_noetherian` states that every fractional ideal of a noetherian integral domain is noetherian
## Implementation notes
Fractional ideals are considered equal when they contain the same elements,
independent of the denominator `a : R` such that `a I ⊆ R`.
Thus, we define `fractional_ideal` to be the subtype of the predicate `is_fractional`,
instead of having `fractional_ideal` be a structure of which `a` is a field.
Most definitions in this file specialize operations from submodules to fractional ideals,
proving that the result of this operation is fractional if the input is fractional.
Exceptions to this rule are defining `(+) := (⊔)` and `⊥ := 0`,
in order to re-use their respective proof terms.
We can still use `simp` to show `↑I + ↑J = ↑(I + J)` and `↑⊥ = ↑0`.
Many results in fact do not need that `P` is a localization, only that `P` is an
`R`-algebra. We omit the `is_localization` parameter whenever this is practical.
Similarly, we don't assume that the localization is a field until we need it to
define ideal quotients. When this assumption is needed, we replace `S` with `R⁰`,
making the localization a field.
## References
* https://en.wikipedia.org/wiki/Fractional_ideal
## Tags
fractional ideal, fractional ideals, invertible ideal
-/
open is_localization
open_locale pointwise
open_locale non_zero_divisors
section defs
variables {R : Type*} [comm_ring R] {S : submonoid R} {P : Type*} [comm_ring P]
variables [algebra R P]
variables (S)
/-- A submodule `I` is a fractional ideal if `a I ⊆ R` for some `a ≠ 0`. -/
def is_fractional (I : submodule R P) :=
∃ a ∈ S, ∀ b ∈ I, is_integer R (a • b)
variables (S P)
/-- The fractional ideals of a domain `R` are ideals of `R` divided by some `a ∈ R`.
More precisely, let `P` be a localization of `R` at some submonoid `S`,
then a fractional ideal `I ⊆ P` is an `R`-submodule of `P`,
such that there is a nonzero `a : R` with `a I ⊆ R`.
-/
def fractional_ideal :=
{I : submodule R P // is_fractional S I}
end defs
namespace fractional_ideal
open set
open submodule
variables {R : Type*} [comm_ring R] {S : submonoid R} {P : Type*} [comm_ring P]
variables [algebra R P] [loc : is_localization S P]
/-- Map a fractional ideal `I` to a submodule by forgetting that `∃ a, a I ⊆ R`.
This coercion is typically called `coe_to_submodule` in lemma names
(or `coe` when the coercion is clear from the context),
not to be confused with `is_localization.coe_submodule : ideal R → submodule R P`
(which we use to define `coe : ideal R → fractional_ideal S P`).
-/
instance : has_coe (fractional_ideal S P) (submodule R P) := ⟨λ I, I.val⟩
protected lemma is_fractional (I : fractional_ideal S P) :
is_fractional S (I : submodule R P) :=
I.prop
section set_like
instance : set_like (fractional_ideal S P) P :=
{ coe := λ I, ↑(I : submodule R P),
coe_injective' := set_like.coe_injective.comp subtype.coe_injective }
@[simp] lemma mem_coe {I : fractional_ideal S P} {x : P} :
x ∈ (I : submodule R P) ↔ x ∈ I :=
iff.rfl
@[ext] lemma ext {I J : fractional_ideal S P} : (∀ x, x ∈ I ↔ x ∈ J) → I = J := set_like.ext
/-- Copy of a `fractional_ideal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. -/
protected def copy (p : fractional_ideal S P) (s : set P) (hs : s = ↑p) : fractional_ideal S P :=
⟨submodule.copy p s hs, by { convert p.is_fractional, ext, simp only [hs], refl }⟩
@[simp] lemma coe_copy (p : fractional_ideal S P) (s : set P) (hs : s = ↑p) :
↑(p.copy s hs) = s :=
rfl
lemma coe_eq (p : fractional_ideal S P) (s : set P) (hs : s = ↑p) : p.copy s hs = p :=
set_like.coe_injective hs
end set_like
@[simp] lemma val_eq_coe (I : fractional_ideal S P) : I.val = I := rfl
@[simp, norm_cast] lemma coe_mk (I : submodule R P) (hI : is_fractional S I) :
(subtype.mk I hI : submodule R P) = I := rfl
/-! Transfer instances from `submodule R P` to `fractional_ideal S P`. -/
instance (I : fractional_ideal S P) : add_comm_group I := submodule.add_comm_group ↑I
instance (I : fractional_ideal S P) : module R I := submodule.module ↑I
lemma coe_to_submodule_injective :
function.injective (coe : fractional_ideal S P → submodule R P) :=
subtype.coe_injective
lemma coe_to_submodule_inj {I J : fractional_ideal S P} : (I : submodule R P) = J ↔ I = J :=
coe_to_submodule_injective.eq_iff
lemma is_fractional_of_le_one (I : submodule R P) (h : I ≤ 1) : is_fractional S I :=
begin
use [1, S.one_mem],
intros b hb,
rw one_smul,
obtain ⟨b', b'_mem, rfl⟩ := h hb,
exact set.mem_range_self b',
end
lemma is_fractional_of_le {I : submodule R P} {J : fractional_ideal S P} (hIJ : I ≤ J) :
is_fractional S I :=
begin
obtain ⟨a, a_mem, ha⟩ := J.is_fractional,
use [a, a_mem],
intros b b_mem,
exact ha b (hIJ b_mem)
end
/-- Map an ideal `I` to a fractional ideal by forgetting `I` is integral.
This is a bundled version of `is_localization.coe_submodule : ideal R → submodule R P`,
which is not to be confused with the `coe : fractional_ideal S P → submodule R P`,
also called `coe_to_submodule` in theorem names.
This map is available as a ring hom, called `fractional_ideal.coe_ideal_hom`.
-/
-- Is a `coe_t` rather than `coe` to speed up failing inference, see library note [use has_coe_t]
instance : has_coe_t (ideal R) (fractional_ideal S P) :=
⟨λ I, ⟨coe_submodule P I,
is_fractional_of_le_one _ $ by simpa using coe_submodule_mono P (le_top : I ≤ ⊤)⟩⟩
@[simp, norm_cast] lemma coe_coe_ideal (I : ideal R) :
((I : fractional_ideal S P) : submodule R P) = coe_submodule P I := rfl
variables (S)
@[simp] lemma mem_coe_ideal {x : P} {I : ideal R} :
x ∈ (I : fractional_ideal S P) ↔ ∃ x', x' ∈ I ∧ algebra_map R P x' = x :=
mem_coe_submodule _ _
lemma mem_coe_ideal_of_mem {x : R} {I : ideal R} (hx : x ∈ I) :
algebra_map R P x ∈ (I : fractional_ideal S P) :=
(mem_coe_ideal S).mpr ⟨x, hx, rfl⟩
lemma coe_ideal_le_coe_ideal' [is_localization S P] (h : S ≤ non_zero_divisors R)
{I J : ideal R} : (I : fractional_ideal S P) ≤ J ↔ I ≤ J :=
coe_submodule_le_coe_submodule h
@[simp] lemma coe_ideal_le_coe_ideal (K : Type*) [comm_ring K] [algebra R K] [is_fraction_ring R K]
{I J : ideal R} : (I : fractional_ideal R⁰ K) ≤ J ↔ I ≤ J :=
is_fraction_ring.coe_submodule_le_coe_submodule
instance : has_zero (fractional_ideal S P) := ⟨(0 : ideal R)⟩
@[simp] lemma mem_zero_iff {x : P} : x ∈ (0 : fractional_ideal S P) ↔ x = 0 :=
⟨(λ ⟨x', x'_mem_zero, x'_eq_x⟩,
have x'_eq_zero : x' = 0 := x'_mem_zero,
by simp [x'_eq_x.symm, x'_eq_zero]),
(λ hx, ⟨0, rfl, by simp [hx]⟩)⟩
variables {S}
@[simp, norm_cast] lemma coe_zero : ↑(0 : fractional_ideal S P) = (⊥ : submodule R P) :=
submodule.ext $ λ _, mem_zero_iff S
@[simp, norm_cast] lemma coe_ideal_bot : ((⊥ : ideal R) : fractional_ideal S P) = 0 := rfl
variables (P)
include loc
@[simp] lemma exists_mem_to_map_eq {x : R} {I : ideal R} (h : S ≤ non_zero_divisors R) :
(∃ x', x' ∈ I ∧ algebra_map R P x' = algebra_map R P x) ↔ x ∈ I :=
⟨λ ⟨x', hx', eq⟩, is_localization.injective _ h eq ▸ hx', λ h, ⟨x, h, rfl⟩⟩
variables {P}
lemma coe_ideal_injective' (h : S ≤ non_zero_divisors R) :
function.injective (coe : ideal R → fractional_ideal S P) :=
λ _ _ h', ((coe_ideal_le_coe_ideal' S h).mp h'.le).antisymm ((coe_ideal_le_coe_ideal' S h).mp h'.ge)
lemma coe_ideal_inj' (h : S ≤ non_zero_divisors R) {I J : ideal R} :
(I : fractional_ideal S P) = J ↔ I = J :=
(coe_ideal_injective' h).eq_iff
@[simp] lemma coe_ideal_eq_zero' {I : ideal R} (h : S ≤ non_zero_divisors R) :
(I : fractional_ideal S P) = 0 ↔ I = (⊥ : ideal R) :=
coe_ideal_inj' h
lemma coe_ideal_ne_zero' {I : ideal R} (h : S ≤ non_zero_divisors R) :
(I : fractional_ideal S P) ≠ 0 ↔ I ≠ (⊥ : ideal R) :=
not_iff_not.mpr $ coe_ideal_eq_zero' h
omit loc
lemma coe_to_submodule_eq_bot {I : fractional_ideal S P} :
(I : submodule R P) = ⊥ ↔ I = 0 :=
⟨λ h, coe_to_submodule_injective (by simp [h]),
λ h, by simp [h]⟩
lemma coe_to_submodule_ne_bot {I : fractional_ideal S P} :
↑I ≠ (⊥ : submodule R P) ↔ I ≠ 0 :=
not_iff_not.mpr coe_to_submodule_eq_bot
instance : inhabited (fractional_ideal S P) := ⟨0⟩
instance : has_one (fractional_ideal S P) :=
⟨(⊤ : ideal R)⟩
variables (S)
@[simp, norm_cast] lemma coe_ideal_top : ((⊤ : ideal R) : fractional_ideal S P) = 1 := rfl
lemma mem_one_iff {x : P} : x ∈ (1 : fractional_ideal S P) ↔ ∃ x' : R, algebra_map R P x' = x :=
iff.intro (λ ⟨x', _, h⟩, ⟨x', h⟩) (λ ⟨x', h⟩, ⟨x', ⟨⟩, h⟩)
lemma coe_mem_one (x : R) : algebra_map R P x ∈ (1 : fractional_ideal S P) :=
(mem_one_iff S).mpr ⟨x, rfl⟩
lemma one_mem_one : (1 : P) ∈ (1 : fractional_ideal S P) :=
(mem_one_iff S).mpr ⟨1, ring_hom.map_one _⟩
variables {S}
/-- `(1 : fractional_ideal S P)` is defined as the R-submodule `f(R) ≤ P`.
However, this is not definitionally equal to `1 : submodule R P`,
which is proved in the actual `simp` lemma `coe_one`. -/
lemma coe_one_eq_coe_submodule_top :
↑(1 : fractional_ideal S P) = coe_submodule P (⊤ : ideal R) :=
rfl
@[simp, norm_cast] lemma coe_one :
(↑(1 : fractional_ideal S P) : submodule R P) = 1 :=
by rw [coe_one_eq_coe_submodule_top, coe_submodule_top]
section lattice
/-!
### `lattice` section
Defines the order on fractional ideals as inclusion of their underlying sets,
and ports the lattice structure on submodules to fractional ideals.
-/
@[simp] lemma coe_le_coe {I J : fractional_ideal S P} :
(I : submodule R P) ≤ (J : submodule R P) ↔ I ≤ J :=
iff.rfl
lemma zero_le (I : fractional_ideal S P) : 0 ≤ I :=
begin
intros x hx,
convert submodule.zero_mem _,
simpa using hx
end
instance order_bot : order_bot (fractional_ideal S P) :=
{ bot := 0,
bot_le := zero_le }
@[simp] lemma bot_eq_zero : (⊥ : fractional_ideal S P) = 0 :=
rfl
@[simp] lemma le_zero_iff {I : fractional_ideal S P} : I ≤ 0 ↔ I = 0 :=
le_bot_iff
lemma eq_zero_iff {I : fractional_ideal S P} : I = 0 ↔ (∀ x ∈ I, x = (0 : P)) :=
⟨ (λ h x hx, by simpa [h, mem_zero_iff] using hx),
(λ h, le_bot_iff.mp (λ x hx, (mem_zero_iff S).mpr (h x hx))) ⟩
lemma _root_.is_fractional.sup {I J : submodule R P} :
is_fractional S I → is_fractional S J → is_fractional S (I ⊔ J)
| ⟨aI, haI, hI⟩ ⟨aJ, haJ, hJ⟩ := ⟨aI * aJ, S.mul_mem haI haJ, λ b hb, begin
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩,
rw smul_add,
apply is_integer_add,
{ rw [mul_smul, smul_comm],
exact is_integer_smul (hI bI hbI), },
{ rw mul_smul,
exact is_integer_smul (hJ bJ hbJ) }
end⟩
lemma _root_.is_fractional.inf_right {I : submodule R P} :
is_fractional S I → ∀ J, is_fractional S (I ⊓ J)
| ⟨aI, haI, hI⟩ J := ⟨aI, haI, λ b hb, begin
rcases mem_inf.mp hb with ⟨hbI, hbJ⟩,
exact hI b hbI
end⟩
instance : has_inf (fractional_ideal S P) := ⟨λ I J, ⟨I ⊓ J, I.is_fractional.inf_right J⟩⟩
@[simp, norm_cast]
lemma coe_inf (I J : fractional_ideal S P) : ↑(I ⊓ J) = (I ⊓ J : submodule R P) := rfl
instance : has_sup (fractional_ideal S P) := ⟨λ I J, ⟨I ⊔ J, I.is_fractional.sup J.is_fractional⟩⟩
@[norm_cast]
lemma coe_sup (I J : fractional_ideal S P) : ↑(I ⊔ J) = (I ⊔ J : submodule R P) := rfl
instance lattice : lattice (fractional_ideal S P) :=
function.injective.lattice _ subtype.coe_injective coe_sup coe_inf
instance : semilattice_sup (fractional_ideal S P) :=
{ ..fractional_ideal.lattice }
end lattice
section semiring
instance : has_add (fractional_ideal S P) := ⟨(⊔)⟩
@[simp]
lemma sup_eq_add (I J : fractional_ideal S P) : I ⊔ J = I + J := rfl
@[simp, norm_cast]
lemma coe_add (I J : fractional_ideal S P) : (↑(I + J) : submodule R P) = I + J := rfl
@[simp, norm_cast]
lemma coe_ideal_sup (I J : ideal R) : ↑(I ⊔ J) = (I + J : fractional_ideal S P) :=
coe_to_submodule_injective $ coe_submodule_sup _ _ _
lemma _root_.is_fractional.nsmul {I : submodule R P} :
Π n : ℕ, is_fractional S I → is_fractional S (n • I : submodule R P)
| 0 _ := begin
rw [zero_smul],
convert ((0 : ideal R) : fractional_ideal S P).is_fractional,
simp,
end
| (n + 1) h := begin
rw succ_nsmul,
exact h.sup (_root_.is_fractional.nsmul n h)
end
instance : has_smul ℕ (fractional_ideal S P) :=
{ smul := λ n I, ⟨n • I, I.is_fractional.nsmul n⟩}
@[norm_cast]
lemma coe_nsmul (n : ℕ) (I : fractional_ideal S P) : (↑(n • I) : submodule R P) = n • I := rfl
lemma _root_.is_fractional.mul {I J : submodule R P} :
is_fractional S I → is_fractional S J → is_fractional S (I * J : submodule R P)
| ⟨aI, haI, hI⟩ ⟨aJ, haJ, hJ⟩ := ⟨aI * aJ, S.mul_mem haI haJ, λ b hb, begin
apply submodule.mul_induction_on hb,
{ intros m hm n hn,
obtain ⟨n', hn'⟩ := hJ n hn,
rw [mul_smul, mul_comm m, ← smul_mul_assoc, ← hn', ← algebra.smul_def],
apply hI,
exact submodule.smul_mem _ _ hm },
{ intros x y hx hy,
rw smul_add,
apply is_integer_add hx hy },
end⟩
lemma _root_.is_fractional.pow {I : submodule R P} (h : is_fractional S I) :
∀ n : ℕ, is_fractional S (I ^ n : submodule R P)
| 0 := is_fractional_of_le_one _ (pow_zero _).le
| (n + 1) := (pow_succ I n).symm ▸ h.mul (_root_.is_fractional.pow n)
/-- `fractional_ideal.mul` is the product of two fractional ideals,
used to define the `has_mul` instance.
This is only an auxiliary definition: the preferred way of writing `I.mul J` is `I * J`.
Elaborated terms involving `fractional_ideal` tend to grow quite large,
so by making definitions irreducible, we hope to avoid deep unfolds.
-/
@[irreducible]
def mul (I J : fractional_ideal S P) : fractional_ideal S P :=
⟨I * J, I.is_fractional.mul J.is_fractional⟩
local attribute [semireducible] mul
instance : has_mul (fractional_ideal S P) := ⟨λ I J, mul I J⟩
@[simp] lemma mul_eq_mul (I J : fractional_ideal S P) : mul I J = I * J := rfl
@[simp, norm_cast]
lemma coe_mul (I J : fractional_ideal S P) : (↑(I * J) : submodule R P) = I * J := rfl
@[simp, norm_cast]
lemma coe_ideal_mul (I J : ideal R) : (↑(I * J) : fractional_ideal S P) = I * J :=
coe_to_submodule_injective $ coe_submodule_mul _ _ _
lemma mul_left_mono (I : fractional_ideal S P) : monotone ((*) I) :=
λ J J' h, mul_le.mpr (λ x hx y hy, mul_mem_mul hx (h hy))
lemma mul_right_mono (I : fractional_ideal S P) : monotone (λ J, J * I) :=
λ J J' h, mul_le.mpr (λ x hx y hy, mul_mem_mul (h hx) hy)
lemma mul_mem_mul {I J : fractional_ideal S P} {i j : P} (hi : i ∈ I) (hj : j ∈ J) :
i * j ∈ I * J := submodule.mul_mem_mul hi hj
lemma mul_le {I J K : fractional_ideal S P} :
I * J ≤ K ↔ (∀ (i ∈ I) (j ∈ J), i * j ∈ K) :=
submodule.mul_le
instance : has_pow (fractional_ideal S P) ℕ := ⟨λ I n, ⟨I^n, I.is_fractional.pow n⟩⟩
@[simp, norm_cast]
lemma coe_pow (I : fractional_ideal S P) (n : ℕ) : ↑(I ^ n) = (I ^ n : submodule R P) := rfl
@[elab_as_eliminator] protected theorem mul_induction_on
{I J : fractional_ideal S P}
{C : P → Prop} {r : P} (hr : r ∈ I * J)
(hm : ∀ (i ∈ I) (j ∈ J), C (i * j))
(ha : ∀ x y, C x → C y → C (x + y)) : C r :=
submodule.mul_induction_on hr hm ha
instance : has_nat_cast (fractional_ideal S P) := ⟨nat.unary_cast⟩
lemma coe_nat_cast (n : ℕ) : ((n : fractional_ideal S P) : submodule R P) = n :=
show ↑n.unary_cast = ↑n, by induction n; simp [*, nat.unary_cast]
instance : comm_semiring (fractional_ideal S P) :=
function.injective.comm_semiring coe subtype.coe_injective
coe_zero coe_one coe_add coe_mul (λ _ _, coe_nsmul _ _) coe_pow coe_nat_cast
section order
lemma add_le_add_left {I J : fractional_ideal S P} (hIJ : I ≤ J) (J' : fractional_ideal S P) :
J' + I ≤ J' + J :=
sup_le_sup_left hIJ J'
lemma mul_le_mul_left {I J : fractional_ideal S P} (hIJ : I ≤ J) (J' : fractional_ideal S P) :
J' * I ≤ J' * J :=
mul_le.mpr (λ k hk j hj, mul_mem_mul hk (hIJ hj))
lemma le_self_mul_self {I : fractional_ideal S P} (hI: 1 ≤ I) : I ≤ I * I :=
begin
convert mul_left_mono I hI,
exact (mul_one I).symm
end
lemma mul_self_le_self {I : fractional_ideal S P} (hI: I ≤ 1) : I * I ≤ I :=
begin
convert mul_left_mono I hI,
exact (mul_one I).symm
end
lemma coe_ideal_le_one {I : ideal R} : (I : fractional_ideal S P) ≤ 1 :=
λ x hx, let ⟨y, _, hy⟩ := (mem_coe_ideal S).mp hx in (mem_one_iff S).mpr ⟨y, hy⟩
lemma le_one_iff_exists_coe_ideal {J : fractional_ideal S P} :
J ≤ (1 : fractional_ideal S P) ↔ ∃ (I : ideal R), ↑I = J :=
begin
split,
{ intro hJ,
refine ⟨⟨{x : R | algebra_map R P x ∈ J}, _, _, _⟩, _⟩,
{ intros a b ha hb,
rw [mem_set_of_eq, ring_hom.map_add],
exact J.val.add_mem ha hb },
{ rw [mem_set_of_eq, ring_hom.map_zero],
exact J.val.zero_mem },
{ intros c x hx,
rw [smul_eq_mul, mem_set_of_eq, ring_hom.map_mul, ← algebra.smul_def],
exact J.val.smul_mem c hx },
{ ext x,
split,
{ rintros ⟨y, hy, eq_y⟩,
rwa ← eq_y },
{ intro hx,
obtain ⟨y, eq_x⟩ := (mem_one_iff S).mp (hJ hx),
rw ← eq_x at *,
exact ⟨y, hx, rfl⟩ } } },
{ rintro ⟨I, hI⟩,
rw ← hI,
apply coe_ideal_le_one },
end
variables (S P)
/-- `coe_ideal_hom (S : submonoid R) P` is `coe : ideal R → fractional_ideal S P` as a ring hom -/
@[simps]
def coe_ideal_hom : ideal R →+* fractional_ideal S P :=
{ to_fun := coe,
map_add' := coe_ideal_sup,
map_mul' := coe_ideal_mul,
map_one' := by rw [ideal.one_eq_top, coe_ideal_top],
map_zero' := coe_ideal_bot }
lemma coe_ideal_pow (I : ideal R) (n : ℕ) : (↑(I^n) : fractional_ideal S P) = I^n :=
(coe_ideal_hom S P).map_pow _ n
open_locale big_operators
lemma coe_ideal_finprod [is_localization S P] {α : Sort*} {f : α → ideal R}
(hS : S ≤ non_zero_divisors R) :
((∏ᶠ a : α, f a : ideal R) : fractional_ideal S P) = ∏ᶠ a : α, (f a : fractional_ideal S P) :=
monoid_hom.map_finprod_of_injective (coe_ideal_hom S P).to_monoid_hom (coe_ideal_injective' hS) f
end order
variables {P' : Type*} [comm_ring P'] [algebra R P'] [loc' : is_localization S P']
variables {P'' : Type*} [comm_ring P''] [algebra R P''] [loc'' : is_localization S P'']
lemma _root_.is_fractional.map (g : P →ₐ[R] P') {I : submodule R P} :
is_fractional S I → is_fractional S (submodule.map g.to_linear_map I)
| ⟨a, a_nonzero, hI⟩ := ⟨a, a_nonzero, λ b hb, begin
obtain ⟨b', b'_mem, hb'⟩ := submodule.mem_map.mp hb,
obtain ⟨x, hx⟩ := hI b' b'_mem,
use x,
erw [←g.commutes, hx, g.map_smul, hb']
end⟩
/-- `I.map g` is the pushforward of the fractional ideal `I` along the algebra morphism `g` -/
def map (g : P →ₐ[R] P') :
fractional_ideal S P → fractional_ideal S P' :=
λ I, ⟨submodule.map g.to_linear_map I, I.is_fractional.map g⟩
@[simp, norm_cast] lemma coe_map (g : P →ₐ[R] P') (I : fractional_ideal S P) :
↑(map g I) = submodule.map g.to_linear_map I := rfl
@[simp] lemma mem_map {I : fractional_ideal S P} {g : P →ₐ[R] P'}
{y : P'} : y ∈ I.map g ↔ ∃ x, x ∈ I ∧ g x = y :=
submodule.mem_map
variables (I J : fractional_ideal S P) (g : P →ₐ[R] P')
@[simp] lemma map_id : I.map (alg_hom.id _ _) = I :=
coe_to_submodule_injective (submodule.map_id I)
@[simp] lemma map_comp (g' : P' →ₐ[R] P'') :
I.map (g'.comp g) = (I.map g).map g' :=
coe_to_submodule_injective (submodule.map_comp g.to_linear_map g'.to_linear_map I)
@[simp, norm_cast] lemma map_coe_ideal (I : ideal R) :
(I : fractional_ideal S P).map g = I :=
begin
ext x,
simp only [mem_coe_ideal],
split,
{ rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩,
exact ⟨y, hy, (g.commutes y).symm⟩ },
{ rintro ⟨y, hy, rfl⟩,
exact ⟨_, ⟨y, hy, rfl⟩, g.commutes y⟩ },
end
@[simp] lemma map_one :
(1 : fractional_ideal S P).map g = 1 :=
map_coe_ideal g ⊤
@[simp] lemma map_zero :
(0 : fractional_ideal S P).map g = 0 :=
map_coe_ideal g 0
@[simp] lemma map_add : (I + J).map g = I.map g + J.map g :=
coe_to_submodule_injective (submodule.map_sup _ _ _)
@[simp] lemma map_mul : (I * J).map g = I.map g * J.map g :=
coe_to_submodule_injective (submodule.map_mul _ _ _)
@[simp] lemma map_map_symm (g : P ≃ₐ[R] P') :
(I.map (g : P →ₐ[R] P')).map (g.symm : P' →ₐ[R] P) = I :=
by rw [←map_comp, g.symm_comp, map_id]
@[simp] lemma map_symm_map (I : fractional_ideal S P') (g : P ≃ₐ[R] P') :
(I.map (g.symm : P' →ₐ[R] P)).map (g : P →ₐ[R] P') = I :=
by rw [←map_comp, g.comp_symm, map_id]
lemma map_mem_map {f : P →ₐ[R] P'} (h : function.injective f) {x : P} {I : fractional_ideal S P} :
f x ∈ map f I ↔ x ∈ I :=
mem_map.trans ⟨λ ⟨x', hx', x'_eq⟩, h x'_eq ▸ hx', λ h, ⟨x, h, rfl⟩⟩
lemma map_injective (f : P →ₐ[R] P') (h : function.injective f) :
function.injective (map f : fractional_ideal S P → fractional_ideal S P') :=
λ I J hIJ, ext (λ x, (map_mem_map h).symm.trans (hIJ.symm ▸ map_mem_map h))
/-- If `g` is an equivalence, `map g` is an isomorphism -/
def map_equiv (g : P ≃ₐ[R] P') :
fractional_ideal S P ≃+* fractional_ideal S P' :=
{ to_fun := map g,
inv_fun := map g.symm,
map_add' := λ I J, map_add I J _,
map_mul' := λ I J, map_mul I J _,
left_inv := λ I, by { rw [←map_comp, alg_equiv.symm_comp, map_id] },
right_inv := λ I, by { rw [←map_comp, alg_equiv.comp_symm, map_id] } }
@[simp] lemma coe_fun_map_equiv (g : P ≃ₐ[R] P') :
(map_equiv g : fractional_ideal S P → fractional_ideal S P') = map g :=
rfl
@[simp] lemma map_equiv_apply (g : P ≃ₐ[R] P') (I : fractional_ideal S P) :
map_equiv g I = map ↑g I := rfl
@[simp] lemma map_equiv_symm (g : P ≃ₐ[R] P') :
((map_equiv g).symm : fractional_ideal S P' ≃+* _) = map_equiv g.symm := rfl
@[simp] lemma map_equiv_refl :
map_equiv alg_equiv.refl = ring_equiv.refl (fractional_ideal S P) :=
ring_equiv.ext (λ x, by simp)
lemma is_fractional_span_iff {s : set P} :
is_fractional S (span R s) ↔ ∃ a ∈ S, ∀ (b : P), b ∈ s → is_integer R (a • b) :=
⟨λ ⟨a, a_mem, h⟩, ⟨a, a_mem, λ b hb, h b (subset_span hb)⟩,
λ ⟨a, a_mem, h⟩, ⟨a, a_mem, λ b hb, span_induction hb
h
(by { rw smul_zero, exact is_integer_zero })
(λ x y hx hy, by { rw smul_add, exact is_integer_add hx hy })
(λ s x hx, by { rw smul_comm, exact is_integer_smul hx })⟩⟩
include loc
lemma is_fractional_of_fg {I : submodule R P} (hI : I.fg) :
is_fractional S I :=
begin
rcases hI with ⟨I, rfl⟩,
rcases exist_integer_multiples_of_finset S I with ⟨⟨s, hs1⟩, hs⟩,
rw is_fractional_span_iff,
exact ⟨s, hs1, hs⟩,
end
omit loc
lemma mem_span_mul_finite_of_mem_mul {I J : fractional_ideal S P} {x : P} (hx : x ∈ I * J) :
∃ (T T' : finset P), (T : set P) ⊆ I ∧ (T' : set P) ⊆ J ∧ x ∈ span R (T * T' : set P) :=
submodule.mem_span_mul_finite_of_mem_mul (by simpa using mem_coe.mpr hx)
variables (S)
lemma coe_ideal_fg (inj : function.injective (algebra_map R P)) (I : ideal R) :
fg ((I : fractional_ideal S P) : submodule R P) ↔ I.fg :=
coe_submodule_fg _ inj _
variables {S}
lemma fg_unit (I : (fractional_ideal S P)ˣ) :
fg (I : submodule R P) :=
begin
have : (1 : P) ∈ (I * ↑I⁻¹ : fractional_ideal S P),
{ rw units.mul_inv, exact one_mem_one _ },
obtain ⟨T, T', hT, hT', one_mem⟩ := mem_span_mul_finite_of_mem_mul this,
refine ⟨T, submodule.span_eq_of_le _ hT _⟩,
rw [← one_mul ↑I, ← mul_one (span R ↑T)],
conv_rhs { rw [← coe_one, ← units.mul_inv I, coe_mul, mul_comm ↑↑I, ← mul_assoc] },
refine submodule.mul_le_mul_left
(le_trans _ (submodule.mul_le_mul_right (submodule.span_le.mpr hT'))),
rwa [submodule.one_le, submodule.span_mul_span]
end
lemma fg_of_is_unit (I : fractional_ideal S P) (h : is_unit I) :
fg (I : submodule R P) :=
by { rcases h with ⟨I, rfl⟩, exact fg_unit I }
lemma _root_.ideal.fg_of_is_unit (inj : function.injective (algebra_map R P))
(I : ideal R) (h : is_unit (I : fractional_ideal S P)) :
I.fg :=
by { rw ← coe_ideal_fg S inj I, exact fg_of_is_unit I h }
variables (S P P')
include loc loc'
/-- `canonical_equiv f f'` is the canonical equivalence between the fractional
ideals in `P` and in `P'` -/
@[irreducible]
noncomputable def canonical_equiv :
fractional_ideal S P ≃+* fractional_ideal S P' :=
map_equiv
{ commutes' := λ r, ring_equiv_of_ring_equiv_eq _ _,
..ring_equiv_of_ring_equiv P P' (ring_equiv.refl R)
(show S.map _ = S, by rw [ring_equiv.to_monoid_hom_refl, submonoid.map_id]) }
@[simp] lemma mem_canonical_equiv_apply {I : fractional_ideal S P} {x : P'} :
x ∈ canonical_equiv S P P' I ↔
∃ y ∈ I, is_localization.map P' (ring_hom.id R)
(λ y (hy : y ∈ S), show ring_hom.id R y ∈ S, from hy) (y : P) = x :=
begin
rw [canonical_equiv, map_equiv_apply, mem_map],
exact ⟨λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩, λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩⟩
end
@[simp] lemma canonical_equiv_symm :
(canonical_equiv S P P').symm = canonical_equiv S P' P :=
ring_equiv.ext $ λ I, set_like.ext_iff.mpr $ λ x,
by { rw [mem_canonical_equiv_apply, canonical_equiv, map_equiv_symm, map_equiv,
ring_equiv.coe_mk, mem_map],
exact ⟨λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩, λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩⟩ }
lemma canonical_equiv_flip (I) :
canonical_equiv S P P' (canonical_equiv S P' P I) = I :=
by rw [←canonical_equiv_symm, ring_equiv.symm_apply_apply]
@[simp]
lemma canonical_equiv_canonical_equiv (P'' : Type*) [comm_ring P''] [algebra R P'']
[is_localization S P''] (I : fractional_ideal S P) :
canonical_equiv S P' P'' (canonical_equiv S P P' I) = canonical_equiv S P P'' I :=
begin
ext,
simp only [is_localization.map_map, ring_hom_inv_pair.comp_eq₂, mem_canonical_equiv_apply,
exists_prop, exists_exists_and_eq_and],
refl
end
lemma canonical_equiv_trans_canonical_equiv (P'' : Type*) [comm_ring P'']
[algebra R P''] [is_localization S P''] :
(canonical_equiv S P P').trans (canonical_equiv S P' P'') = canonical_equiv S P P'' :=
ring_equiv.ext (canonical_equiv_canonical_equiv S P P' P'')
@[simp]
lemma canonical_equiv_coe_ideal (I : ideal R) :
canonical_equiv S P P' I = I :=
by { ext, simp [is_localization.map_eq] }
omit loc'
@[simp]
lemma canonical_equiv_self : canonical_equiv S P P = ring_equiv.refl _ :=
begin
rw ← canonical_equiv_trans_canonical_equiv S P P,
convert (canonical_equiv S P P).symm_trans_self,
exact (canonical_equiv_symm S P P).symm
end
end semiring
section is_fraction_ring
/-!
### `is_fraction_ring` section
This section concerns fractional ideals in the field of fractions,
i.e. the type `fractional_ideal R⁰ K` where `is_fraction_ring R K`.
-/
variables {K K' : Type*} [field K] [field K']
variables [algebra R K] [is_fraction_ring R K] [algebra R K'] [is_fraction_ring R K']
variables {I J : fractional_ideal R⁰ K} (h : K →ₐ[R] K')
/-- Nonzero fractional ideals contain a nonzero integer. -/
lemma exists_ne_zero_mem_is_integer [nontrivial R] (hI : I ≠ 0) :
∃ x ≠ (0 : R), algebra_map R K x ∈ I :=
begin
obtain ⟨y, y_mem, y_not_mem⟩ := set_like.exists_of_lt
(by simpa only using bot_lt_iff_ne_bot.mpr hI),
have y_ne_zero : y ≠ 0 := by simpa using y_not_mem,
obtain ⟨z, ⟨x, hx⟩⟩ := exists_integer_multiple R⁰ y,
refine ⟨x, _, _⟩,
{ rw [ne.def, ← @is_fraction_ring.to_map_eq_zero_iff R _ K, hx, algebra.smul_def],
exact mul_ne_zero (is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors z.2) y_ne_zero },
{ rw hx,
exact smul_mem _ _ y_mem }
end
lemma map_ne_zero [nontrivial R] (hI : I ≠ 0) : I.map h ≠ 0 :=
begin
obtain ⟨x, x_ne_zero, hx⟩ := exists_ne_zero_mem_is_integer hI,
contrapose! x_ne_zero with map_eq_zero,
refine is_fraction_ring.to_map_eq_zero_iff.mp (eq_zero_iff.mp map_eq_zero _ (mem_map.mpr _)),
exact ⟨algebra_map R K x, hx, h.commutes x⟩,
end
@[simp] lemma map_eq_zero_iff [nontrivial R] : I.map h = 0 ↔ I = 0 :=
⟨imp_of_not_imp_not _ _ (map_ne_zero _), λ hI, hI.symm ▸ map_zero h⟩
lemma coe_ideal_injective : function.injective (coe : ideal R → fractional_ideal R⁰ K) :=
coe_ideal_injective' le_rfl
lemma coe_ideal_inj {I J : ideal R} :
(I : fractional_ideal R⁰ K) = (J : fractional_ideal R⁰ K) ↔ I = J :=
coe_ideal_inj' le_rfl
@[simp] lemma coe_ideal_eq_zero {I : ideal R} : (I : fractional_ideal R⁰ K) = 0 ↔ I = ⊥ :=
coe_ideal_eq_zero' le_rfl
lemma coe_ideal_ne_zero {I : ideal R} : (I : fractional_ideal R⁰ K) ≠ 0 ↔ I ≠ ⊥ :=
coe_ideal_ne_zero' le_rfl
@[simp] lemma coe_ideal_eq_one {I : ideal R} : (I : fractional_ideal R⁰ K) = 1 ↔ I = 1 :=
by simpa only [ideal.one_eq_top] using coe_ideal_inj
lemma coe_ideal_ne_one {I : ideal R} : (I : fractional_ideal R⁰ K) ≠ 1 ↔ I ≠ 1 :=
not_iff_not.mpr coe_ideal_eq_one
end is_fraction_ring
section quotient
/-!
### `quotient` section
This section defines the ideal quotient of fractional ideals.
In this section we need that each non-zero `y : R` has an inverse in
the localization, i.e. that the localization is a field. We satisfy this
assumption by taking `S = non_zero_divisors R`, `R`'s localization at which
is a field because `R` is a domain.
-/
open_locale classical
variables {R₁ : Type*} [comm_ring R₁] {K : Type*} [field K]
variables [algebra R₁ K] [frac : is_fraction_ring R₁ K]
instance : nontrivial (fractional_ideal R₁⁰ K) :=
⟨⟨0, 1, λ h,
have this : (1 : K) ∈ (0 : fractional_ideal R₁⁰ K) :=
by { rw ← (algebra_map R₁ K).map_one, simpa only [h] using coe_mem_one R₁⁰ 1 },
one_ne_zero ((mem_zero_iff _).mp this)⟩⟩
lemma ne_zero_of_mul_eq_one (I J : fractional_ideal R₁⁰ K) (h : I * J = 1) : I ≠ 0 :=
λ hI, zero_ne_one' (fractional_ideal R₁⁰ K) (by { convert h, simp [hI], })
variables [is_domain R₁]
include frac
lemma _root_.is_fractional.div_of_nonzero {I J : submodule R₁ K} :
is_fractional R₁⁰ I → is_fractional R₁⁰ J → J ≠ 0 → is_fractional R₁⁰ (I / J)
| ⟨aI, haI, hI⟩ ⟨aJ, haJ, hJ⟩ h := begin
obtain ⟨y, mem_J, not_mem_zero⟩ := set_like.exists_of_lt
(by simpa only using bot_lt_iff_ne_bot.mpr h),
obtain ⟨y', hy'⟩ := hJ y mem_J,
use (aI * y'),
split,
{ apply (non_zero_divisors R₁).mul_mem haI (mem_non_zero_divisors_iff_ne_zero.mpr _),
intro y'_eq_zero,
have : algebra_map R₁ K aJ * y = 0,
{ rw [← algebra.smul_def, ←hy', y'_eq_zero, ring_hom.map_zero] },
have y_zero := (mul_eq_zero.mp this).resolve_left
(mt ((injective_iff_map_eq_zero (algebra_map R₁ K)).1 (is_fraction_ring.injective _ _) _)
(mem_non_zero_divisors_iff_ne_zero.mp haJ)),
apply not_mem_zero,
simpa only using (mem_zero_iff R₁⁰).mpr y_zero, },
intros b hb,
convert hI _ (hb _ (submodule.smul_mem _ aJ mem_J)) using 1,
rw [← hy', mul_comm b, ← algebra.smul_def, mul_smul]
end
lemma fractional_div_of_nonzero {I J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :
is_fractional R₁⁰ (I / J : submodule R₁ K) :=
I.is_fractional.div_of_nonzero J.is_fractional $ λ H, h $
coe_to_submodule_injective $ H.trans coe_zero.symm
noncomputable instance fractional_ideal_has_div :
has_div (fractional_ideal R₁⁰ K) :=
⟨ λ I J, if h : J = 0 then 0 else ⟨I / J, fractional_div_of_nonzero h⟩ ⟩
variables {I J : fractional_ideal R₁⁰ K} [ J ≠ 0 ]
@[simp] lemma div_zero {I : fractional_ideal R₁⁰ K} :
I / 0 = 0 :=
dif_pos rfl
lemma div_nonzero {I J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :
(I / J) = ⟨I / J, fractional_div_of_nonzero h⟩ :=
dif_neg h
@[simp] lemma coe_div {I J : fractional_ideal R₁⁰ K} (hJ : J ≠ 0) :
(↑(I / J) : submodule R₁ K) = ↑I / (↑J : submodule R₁ K) :=
congr_arg _ (dif_neg hJ)
lemma mem_div_iff_of_nonzero {I J : fractional_ideal R₁⁰ K} (h : J ≠ 0) {x} :
x ∈ I / J ↔ ∀ y ∈ J, x * y ∈ I :=
by { rw div_nonzero h, exact submodule.mem_div_iff_forall_mul_mem }
lemma mul_one_div_le_one {I : fractional_ideal R₁⁰ K} : I * (1 / I) ≤ 1 :=
begin
by_cases hI : I = 0,
{ rw [hI, div_zero, mul_zero],
exact zero_le 1 },
{ rw [← coe_le_coe, coe_mul, coe_div hI, coe_one],
apply submodule.mul_one_div_le_one },
end
lemma le_self_mul_one_div {I : fractional_ideal R₁⁰ K} (hI : I ≤ (1 : fractional_ideal R₁⁰ K)) :
I ≤ I * (1 / I) :=
begin
by_cases hI_nz : I = 0,
{ rw [hI_nz, div_zero, mul_zero], exact zero_le 0 },
{ rw [← coe_le_coe, coe_mul, coe_div hI_nz, coe_one],
rw [← coe_le_coe, coe_one] at hI,
exact submodule.le_self_mul_one_div hI },
end
lemma le_div_iff_of_nonzero {I J J' : fractional_ideal R₁⁰ K} (hJ' : J' ≠ 0) :
I ≤ J / J' ↔ ∀ (x ∈ I) (y ∈ J'), x * y ∈ J :=
⟨ λ h x hx, (mem_div_iff_of_nonzero hJ').mp (h hx),
λ h x hx, (mem_div_iff_of_nonzero hJ').mpr (h x hx) ⟩
lemma le_div_iff_mul_le {I J J' : fractional_ideal R₁⁰ K} (hJ' : J' ≠ 0) :
I ≤ J / J' ↔ I * J' ≤ J :=
begin
rw div_nonzero hJ',
convert submodule.le_div_iff_mul_le using 1,
rw [← coe_mul, coe_le_coe]
end
@[simp] lemma div_one {I : fractional_ideal R₁⁰ K} : I / 1 = I :=
begin
rw [div_nonzero (one_ne_zero' (fractional_ideal R₁⁰ K))],
ext,
split; intro h,
{ simpa using mem_div_iff_forall_mul_mem.mp h 1
((algebra_map R₁ K).map_one ▸ coe_mem_one R₁⁰ 1) },
{ apply mem_div_iff_forall_mul_mem.mpr,
rintros y ⟨y', _, rfl⟩,
rw mul_comm,
convert submodule.smul_mem _ y' h,
exact (algebra.smul_def _ _).symm }
end
theorem eq_one_div_of_mul_eq_one_right (I J : fractional_ideal R₁⁰ K) (h : I * J = 1) :
J = 1 / I :=
begin
have hI : I ≠ 0 := ne_zero_of_mul_eq_one I J h,
suffices h' : I * (1 / I) = 1,
{ exact (congr_arg units.inv $
@units.ext _ _ (units.mk_of_mul_eq_one _ _ h) (units.mk_of_mul_eq_one _ _ h') rfl) },
apply le_antisymm,
{ apply mul_le.mpr _,
intros x hx y hy,
rw mul_comm,
exact (mem_div_iff_of_nonzero hI).mp hy x hx },
rw ← h,
apply mul_left_mono I,
apply (le_div_iff_of_nonzero hI).mpr _,
intros y hy x hx,
rw mul_comm,
exact mul_mem_mul hx hy,
end
theorem mul_div_self_cancel_iff {I : fractional_ideal R₁⁰ K} :
I * (1 / I) = 1 ↔ ∃ J, I * J = 1 :=
⟨λ h, ⟨(1 / I), h⟩, λ ⟨J, hJ⟩, by rwa [← eq_one_div_of_mul_eq_one_right I J hJ]⟩
variables {K' : Type*} [field K'] [algebra R₁ K'] [is_fraction_ring R₁ K']
@[simp] lemma map_div (I J : fractional_ideal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(I / J).map (h : K →ₐ[R₁] K') = I.map h / J.map h :=
begin
by_cases H : J = 0,
{ rw [H, div_zero, map_zero, div_zero] },
{ apply coe_to_submodule_injective,
simp [div_nonzero H, div_nonzero (map_ne_zero _ H), submodule.map_div] }
end
@[simp] lemma map_one_div (I : fractional_ideal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(1 / I).map (h : K →ₐ[R₁] K') = 1 / I.map h :=
by rw [map_div, map_one]
end quotient