statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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is_fraction_ring_iff_of_base_ring_equiv (h : R ≃+* P) :
is_fraction_ring R S ↔
@@is_fraction_ring P _ S _ ((algebra_map R S).comp h.symm.to_ring_hom).to_algebra | begin
delta is_fraction_ring,
convert is_localization_iff_of_base_ring_equiv _ _ h,
ext x,
erw submonoid.map_equiv_eq_comap_symm,
simp only [mul_equiv.coe_to_monoid_hom,
ring_equiv.to_mul_equiv_eq_coe, submonoid.mem_comap],
split,
{ rintros hx z (hz : z * h.symm x = 0),
rw ← h.map_eq_zero_iff,
... | lemma | is_fraction_ring.is_fraction_ring_iff_of_base_ring_equiv | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map",
"is_fraction_ring",
"mul_equiv.coe_to_monoid_hom",
"ring_equiv.to_mul_equiv_eq_coe",
"submonoid.map_equiv_eq_comap_symm",
"submonoid.mem_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial (R S : Type*) [comm_ring R] [nontrivial R] [comm_ring S] [algebra R S]
[is_fraction_ring R S] : nontrivial S | begin
apply nontrivial_of_ne,
intro h,
apply @zero_ne_one R,
exact is_localization.injective S (le_of_eq rfl)
(((algebra_map R S).map_zero.trans h).trans (algebra_map R S).map_one.symm),
end | lemma | is_fraction_ring.nontrivial | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra",
"algebra_map",
"comm_ring",
"is_fraction_ring",
"is_localization.injective",
"nontrivial",
"nontrivial_of_ne",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fraction_ring | localization (non_zero_divisors R) | def | fraction_ring | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"localization",
"non_zero_divisors"
] | The fraction ring of a commutative ring `R` as a quotient type.
We instantiate this definition as generally as possible, and assume that the
commutative ring `R` is an integral domain only when this is needed for proving. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique [subsingleton R] : unique (fraction_ring R) | localization.unique | instance | fraction_ring.unique | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"fraction_ring",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_div {r s} : (localization.mk r s : fraction_ring A) =
(algebra_map _ _ r / algebra_map A _ s : fraction_ring A) | by rw [localization.mk_eq_mk', is_fraction_ring.mk'_eq_div] | lemma | fraction_ring.mk_eq_div | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map",
"fraction_ring",
"is_fraction_ring.mk'_eq_div",
"localization.mk",
"localization.mk_eq_mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_equiv (K : Type*) [field K] [algebra A K] [is_fraction_ring A K] :
fraction_ring A ≃ₐ[A] K | localization.alg_equiv (non_zero_divisors A) K | def | fraction_ring.alg_equiv | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"alg_equiv",
"algebra",
"field",
"fraction_ring",
"is_fraction_ring",
"localization.alg_equiv",
"non_zero_divisors"
] | Given an integral domain `A` and a localization map to a field of fractions
`f : A →+* K`, we get an `A`-isomorphism between the field of fractions of `A` as a quotient
type and `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_ideal (I : ideal R) : ideal S | { carrier := { z : S | ∃ x : I × M, z * algebra_map R S x.2 = algebra_map R S x.1},
zero_mem' := ⟨⟨0, 1⟩, by simp⟩,
add_mem' := begin
rintros a b ⟨a', ha⟩ ⟨b', hb⟩,
use ⟨a'.2 * b'.1 + b'.2 * a'.1, I.add_mem (I.mul_mem_left _ b'.1.2) (I.mul_mem_left _ a'.1.2)⟩,
use a'.2 * b'.2,
simp only [ring_hom.ma... | def | is_localization.map_ideal | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"ideal",
"mul_assoc",
"mul_comm",
"ring",
"ring_hom.map_add",
"ring_hom.map_mul",
"smul_eq_mul",
"submodule.coe_mk",
"submonoid.coe_mul"
] | Explicit characterization of the ideal given by `ideal.map (algebra_map R S) I`.
In practice, this ideal differs only in that the carrier set is defined explicitly.
This definition is only meant to be used in proving `mem_map_algebra_map_iff`,
and any proof that needs to refer to the explicit carrier set should use tha... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_map_algebra_map_iff {I : ideal R} {z} :
z ∈ ideal.map (algebra_map R S) I ↔ ∃ x : I × M, z * algebra_map R S x.2 = algebra_map R S x.1 | begin
split,
{ change _ → z ∈ map_ideal M S I,
refine λ h, ideal.mem_Inf.1 h (λ z hz, _),
obtain ⟨y, hy⟩ := hz,
use ⟨⟨⟨y, hy.left⟩, 1⟩, by simp [hy.right]⟩ },
{ rintros ⟨⟨a, s⟩, h⟩,
rw [← ideal.unit_mul_mem_iff_mem _ (map_units S s), mul_comm],
exact h.symm ▸ ideal.mem_map_of_mem _ a.2 }
end | theorem | is_localization.mem_map_algebra_map_iff | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"ideal",
"ideal.map",
"ideal.mem_map_of_mem",
"ideal.unit_mul_mem_iff_mem",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap (J : ideal S) :
ideal.map (algebra_map R S) (ideal.comap (algebra_map R S) J) = J | le_antisymm (ideal.map_le_iff_le_comap.2 le_rfl) $ λ x hJ,
begin
obtain ⟨r, s, hx⟩ := mk'_surjective M x,
rw ←hx at ⊢ hJ,
exact ideal.mul_mem_right _ _ (ideal.mem_map_of_mem _ (show (algebra_map R S) r ∈ J, from
mk'_spec S r s ▸ J.mul_mem_right ((algebra_map R S) s) hJ)),
end | theorem | is_localization.map_comap | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"ideal",
"ideal.comap",
"ideal.map",
"ideal.mem_map_of_mem",
"ideal.mul_mem_right",
"le_rfl",
"mk'_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_map_of_is_prime_disjoint (I : ideal R) (hI : I.is_prime)
(hM : disjoint (M : set R) I) :
ideal.comap (algebra_map R S) (ideal.map (algebra_map R S) I) = I | begin
refine le_antisymm (λ a ha, _) ideal.le_comap_map,
obtain ⟨⟨b, s⟩, h⟩ := (mem_map_algebra_map_iff M S).1 (ideal.mem_comap.1 ha),
replace h : algebra_map R S (s * a) = algebra_map R S b :=
by simpa only [←map_mul, mul_comm] using h,
obtain ⟨c, hc⟩ := (eq_iff_exists M S).1 h,
have : (↑c * ↑s) * a ∈ I ... | theorem | is_localization.comap_map_of_is_prime_disjoint | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"disjoint",
"ideal",
"ideal.comap",
"ideal.le_comap_map",
"ideal.map",
"mul_assoc",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_embedding : ideal S ↪o ideal R | { to_fun := λ J, ideal.comap (algebra_map R S) J,
inj' := function.left_inverse.injective (map_comap M S),
map_rel_iff' := λ J₁ J₂, ⟨λ hJ, (map_comap M S) J₁ ▸ (map_comap M S) J₂ ▸ ideal.map_mono hJ,
ideal.comap_mono⟩ } | def | is_localization.order_embedding | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"ideal",
"ideal.comap",
"ideal.map_mono",
"order_embedding"
] | If `S` is the localization of `R` at a submonoid, the ordering of ideals of `S` is
embedded in the ordering of ideals of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_prime_iff_is_prime_disjoint (J : ideal S) :
J.is_prime ↔ (ideal.comap (algebra_map R S) J).is_prime ∧
disjoint (M : set R) ↑(ideal.comap (algebra_map R S) J) | begin
split,
{ refine λ h, ⟨⟨_, _⟩, set.disjoint_left.mpr $ λ m hm1 hm2,
h.ne_top (ideal.eq_top_of_is_unit_mem _ hm2 (map_units S ⟨m, hm1⟩))⟩,
{ refine λ hJ, h.ne_top _,
rw [eq_top_iff, ← (order_embedding M S).le_iff_le],
exact le_of_eq hJ.symm },
{ intros x y hxy,
rw [ideal.mem_coma... | lemma | is_localization.is_prime_iff_is_prime_disjoint | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"disjoint",
"eq_top_iff",
"ideal",
"ideal.comap",
"ideal.eq_top_of_is_unit_mem",
"ideal.mem_comap",
"mk'",
"mk'_surjective",
"order_embedding",
"ring_hom.map_mul"
] | If `R` is a ring, then prime ideals in the localization at `M`
correspond to prime ideals in the original ring `R` that are disjoint from `M`.
This lemma gives the particular case for an ideal and its comap,
see `le_rel_iso_of_prime` for the more general relation isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_prime_of_is_prime_disjoint (I : ideal R) (hp : I.is_prime)
(hd : disjoint (M : set R) ↑I) : (ideal.map (algebra_map R S) I).is_prime | begin
rw [is_prime_iff_is_prime_disjoint M S, comap_map_of_is_prime_disjoint M S I hp hd],
exact ⟨hp, hd⟩
end | lemma | is_localization.is_prime_of_is_prime_disjoint | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"disjoint",
"ideal",
"ideal.map"
] | If `R` is a ring, then prime ideals in the localization at `M`
correspond to prime ideals in the original ring `R` that are disjoint from `M`.
This lemma gives the particular case for an ideal and its map,
see `le_rel_iso_of_prime` for the more general relation isomorphism, and the reverse implication | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso_of_prime :
{p : ideal S // p.is_prime} ≃o {p : ideal R // p.is_prime ∧ disjoint (M : set R) ↑p} | { to_fun := λ p, ⟨ideal.comap (algebra_map R S) p.1,
(is_prime_iff_is_prime_disjoint M S p.1).1 p.2⟩,
inv_fun := λ p, ⟨ideal.map (algebra_map R S) p.1,
is_prime_of_is_prime_disjoint M S p.1 p.2.1 p.2.2⟩,
left_inv := λ J, subtype.eq (map_comap M S J),
right_inv := λ I, subtype.... | def | is_localization.order_iso_of_prime | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"disjoint",
"ideal",
"ideal.map_mono",
"inv_fun"
] | If `R` is a ring, then prime ideals in the localization at `M`
correspond to prime ideals in the original ring `R` that are disjoint from `M` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective_quotient_map_of_maximal_of_localization {I : ideal S} [I.is_prime] {J : ideal R}
{H : J ≤ I.comap (algebra_map R S)} (hI : (I.comap (algebra_map R S)).is_maximal) :
function.surjective (I.quotient_map (algebra_map R S) H) | begin
intro s,
obtain ⟨s, rfl⟩ := ideal.quotient.mk_surjective s,
obtain ⟨r, ⟨m, hm⟩, rfl⟩ := mk'_surjective M s,
by_cases hM : (ideal.quotient.mk (I.comap (algebra_map R S))) m = 0,
{ have : I = ⊤,
{ rw ideal.eq_top_iff_one,
rw [ideal.quotient.eq_zero_iff_mem, ideal.mem_comap] at hM,
convert ... | lemma | is_localization.surjective_quotient_map_of_maximal_of_localization | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"ideal",
"ideal.eq_top_iff_one",
"ideal.mem_comap",
"ideal.quotient.eq_zero_iff_mem",
"ideal.quotient.maximal_ideal_iff_is_field_quotient",
"ideal.quotient.mk",
"ideal.quotient.mk_surjective",
"ideal.quotient_map",
"ideal.quotient_map_mk",
"le_rfl",
"mk'",
"mk'_surjective",
... | `quotient_map` applied to maximal ideals of a localization is `surjective`.
The quotient by a maximal ideal is a field, so inverses to elements already exist,
and the localization necessarily maps the equivalence class of the inverse in the localization | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bot_lt_comap_prime [is_domain R] (hM : M ≤ R⁰)
(p : ideal S) [hpp : p.is_prime] (hp0 : p ≠ ⊥) :
⊥ < ideal.comap (algebra_map R S) p | begin
haveI : is_domain S := is_domain_of_le_non_zero_divisors _ hM,
convert (order_iso_of_prime M S).lt_iff_lt.mpr
(show (⟨⊥, ideal.bot_prime⟩ : {p : ideal S // p.is_prime}) < ⟨p, hpp⟩, from hp0.bot_lt),
exact (ideal.comap_bot_of_injective (algebra_map R S) (is_localization.injective _ hM)).symm,
end | lemma | is_localization.bot_lt_comap_prime | ring_theory.localization | src/ring_theory/localization/ideal.lean | [
"ring_theory.ideal.quotient_operations",
"ring_theory.localization.basic"
] | [
"algebra_map",
"ideal",
"ideal.comap",
"ideal.comap_bot_of_injective",
"is_domain",
"is_localization.injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integer (a : S) : Prop | a ∈ (algebra_map R S).range | def | is_localization.is_integer | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"algebra_map"
] | Given `a : S`, `S` a localization of `R`, `is_integer R a` iff `a` is in the image of
the localization map from `R` to `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integer_zero : is_integer R (0 : S) | subring.zero_mem _ | lemma | is_localization.is_integer_zero | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"subring.zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integer_one : is_integer R (1 : S) | subring.one_mem _ | lemma | is_localization.is_integer_one | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"subring.one_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integer_add {a b : S} (ha : is_integer R a) (hb : is_integer R b) :
is_integer R (a + b) | subring.add_mem _ ha hb | lemma | is_localization.is_integer_add | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"subring.add_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integer_mul {a b : S} (ha : is_integer R a) (hb : is_integer R b) :
is_integer R (a * b) | subring.mul_mem _ ha hb | lemma | is_localization.is_integer_mul | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"subring.mul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integer_smul {a : R} {b : S} (hb : is_integer R b) :
is_integer R (a • b) | begin
rcases hb with ⟨b', hb⟩,
use a * b',
rw [←hb, (algebra_map R S).map_mul, algebra.smul_def]
end | lemma | is_localization.is_integer_smul | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"algebra.smul_def",
"algebra_map",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_integer_multiple' (a : S) :
∃ (b : M), is_integer R (a * algebra_map R S b) | let ⟨⟨num, denom⟩, h⟩ := is_localization.surj _ a in ⟨denom, set.mem_range.mpr ⟨num, h.symm⟩⟩ | lemma | is_localization.exists_integer_multiple' | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"algebra_map"
] | Each element `a : S` has an `M`-multiple which is an integer.
This version multiplies `a` on the right, matching the argument order in `localization_map.surj`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_integer_multiple (a : S) :
∃ (b : M), is_integer R ((b : R) • a) | by { simp_rw [algebra.smul_def, mul_comm _ a], apply exists_integer_multiple' } | lemma | is_localization.exists_integer_multiple | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"algebra.smul_def",
"mul_comm"
] | Each element `a : S` has an `M`-multiple which is an integer.
This version multiplies `a` on the left, matching the argument order in the `has_smul` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exist_integer_multiples {ι : Type*} (s : finset ι) (f : ι → S) :
∃ (b : M), ∀ i ∈ s, is_localization.is_integer R ((b : R) • f i) | begin
haveI := classical.prop_decidable,
refine ⟨∏ i in s, (sec M (f i)).2, λ i hi, ⟨_, _⟩⟩,
{ exact (∏ j in s.erase i, (sec M (f j)).2) * (sec M (f i)).1 },
rw [ring_hom.map_mul, sec_spec', ←mul_assoc, ←(algebra_map R S).map_mul, ← algebra.smul_def],
congr' 2,
refine trans _ ((submonoid.subtype M).map_prod... | lemma | is_localization.exist_integer_multiples | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"algebra.smul_def",
"algebra_map",
"finset",
"finset.insert_erase",
"is_localization.is_integer",
"map_mul",
"map_prod",
"mul_comm",
"ring_hom.map_mul",
"submonoid.subtype"
] | We can clear the denominators of a `finset`-indexed family of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exist_integer_multiples_of_finite {ι : Type*} [finite ι] (f : ι → S) :
∃ (b : M), ∀ i, is_localization.is_integer R ((b : R) • f i) | begin
casesI nonempty_fintype ι,
obtain ⟨b, hb⟩ := exist_integer_multiples M finset.univ f,
exact ⟨b, λ i, hb i (finset.mem_univ _)⟩
end | lemma | is_localization.exist_integer_multiples_of_finite | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"finite",
"finset.mem_univ",
"finset.univ",
"is_localization.is_integer",
"nonempty_fintype"
] | We can clear the denominators of a finite indexed family of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exist_integer_multiples_of_finset (s : finset S) :
∃ (b : M), ∀ a ∈ s, is_integer R ((b : R) • a) | exist_integer_multiples M s id | lemma | is_localization.exist_integer_multiples_of_finset | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"finset"
] | We can clear the denominators of a finite set of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
common_denom {ι : Type*} (s : finset ι) (f : ι → S) : M | (exist_integer_multiples M s f).some | def | is_localization.common_denom | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"finset"
] | A choice of a common multiple of the denominators of a `finset`-indexed family of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integer_multiple {ι : Type*} (s : finset ι) (f : ι → S) (i : s) : R | ((exist_integer_multiples M s f).some_spec i i.prop).some | def | is_localization.integer_multiple | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"finset"
] | The numerator of a fraction after clearing the denominators
of a `finset`-indexed family of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_integer_multiple {ι : Type*} (s : finset ι) (f : ι → S) (i : s) :
algebra_map R S (integer_multiple M s f i) = common_denom M s f • f i | ((exist_integer_multiples M s f).some_spec _ i.prop).some_spec | lemma | is_localization.map_integer_multiple | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"algebra_map",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
common_denom_of_finset (s : finset S) : M | common_denom M s id | def | is_localization.common_denom_of_finset | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"finset"
] | A choice of a common multiple of the denominators of a finite set of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finset_integer_multiple [decidable_eq R] (s : finset S) : finset R | s.attach.image (λ t, integer_multiple M s id t) | def | is_localization.finset_integer_multiple | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"finset"
] | The finset of numerators after clearing the denominators of a finite set of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finset_integer_multiple_image [decidable_eq R] (s : finset S) :
algebra_map R S '' (finset_integer_multiple M s) =
common_denom_of_finset M s • s | begin
delta finset_integer_multiple common_denom,
rw finset.coe_image,
ext,
split,
{ rintro ⟨_, ⟨x, -, rfl⟩, rfl⟩,
rw map_integer_multiple,
exact set.mem_image_of_mem _ x.prop },
{ rintro ⟨x, hx, rfl⟩,
exact ⟨_, ⟨⟨x, hx⟩, s.mem_attach _, rfl⟩, map_integer_multiple M s id _⟩ }
end | lemma | is_localization.finset_integer_multiple_image | ring_theory.localization | src/ring_theory/localization/integer.lean | [
"ring_theory.localization.basic"
] | [
"algebra_map",
"finset",
"finset.coe_image",
"set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_integer_normalization (p : S[X]) (i : ℕ) : R | if hi : i ∈ p.support
then classical.some (classical.some_spec
(exist_integer_multiples_of_finset M (p.support.image p.coeff))
(p.coeff i)
(finset.mem_image.mpr ⟨i, hi, rfl⟩))
else 0 | def | is_localization.coeff_integer_normalization | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [] | `coeff_integer_normalization p` gives the coefficients of the polynomial
`integer_normalization p` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coeff_integer_normalization_of_not_mem_support (p : S[X]) (i : ℕ)
(h : coeff p i = 0) : coeff_integer_normalization M p i = 0 | by simp only [coeff_integer_normalization, h, mem_support_iff, eq_self_iff_true, not_true,
ne.def, dif_neg, not_false_iff] | lemma | is_localization.coeff_integer_normalization_of_not_mem_support | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_integer_normalization_mem_support (p : S[X]) (i : ℕ)
(h : coeff_integer_normalization M p i ≠ 0) : i ∈ p.support | begin
contrapose h,
rw [ne.def, not_not, coeff_integer_normalization, dif_neg h]
end | lemma | is_localization.coeff_integer_normalization_mem_support | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integer_normalization (p : S[X]) :
R[X] | ∑ i in p.support, monomial i (coeff_integer_normalization M p i) | def | is_localization.integer_normalization | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [] | `integer_normalization g` normalizes `g` to have integer coefficients
by clearing the denominators | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integer_normalization_coeff (p : S[X]) (i : ℕ) :
(integer_normalization M p).coeff i = coeff_integer_normalization M p i | by simp [integer_normalization, coeff_monomial, coeff_integer_normalization_of_not_mem_support]
{contextual := tt} | lemma | is_localization.integer_normalization_coeff | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integer_normalization_spec (p : S[X]) :
∃ (b : M), ∀ i,
algebra_map R S ((integer_normalization M p).coeff i) = (b : R) • p.coeff i | begin
use classical.some (exist_integer_multiples_of_finset M (p.support.image p.coeff)),
intro i,
rw [integer_normalization_coeff, coeff_integer_normalization],
split_ifs with hi,
{ exact classical.some_spec (classical.some_spec
(exist_integer_multiples_of_finset M (p.support.image p.coeff))
(p.c... | lemma | is_localization.integer_normalization_spec | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"ring_hom.map_zero",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integer_normalization_map_to_map (p : S[X]) :
∃ (b : M), (integer_normalization M p).map (algebra_map R S) = (b : R) • p | let ⟨b, hb⟩ := integer_normalization_spec M p in
⟨b, polynomial.ext (λ i, by { rw [coeff_map, coeff_smul], exact hb i })⟩ | lemma | is_localization.integer_normalization_map_to_map | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"polynomial.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integer_normalization_eval₂_eq_zero (g : S →+* R') (p : S[X])
{x : R'} (hx : eval₂ g x p = 0) :
eval₂ (g.comp (algebra_map R S)) x (integer_normalization M p) = 0 | let ⟨b, hb⟩ := integer_normalization_map_to_map M p in
trans (eval₂_map (algebra_map R S) g x).symm
(by rw [hb, ← is_scalar_tower.algebra_map_smul S (b : R) p, eval₂_smul, hx, mul_zero]) | lemma | is_localization.integer_normalization_eval₂_eq_zero | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"is_scalar_tower.algebra_map_smul",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integer_normalization_aeval_eq_zero [algebra R R'] [algebra S R'] [is_scalar_tower R S R']
(p : S[X]) {x : R'} (hx : aeval x p = 0) :
aeval x (integer_normalization M p) = 0 | by rw [aeval_def, is_scalar_tower.algebra_map_eq R S R',
integer_normalization_eval₂_eq_zero _ _ _ hx] | lemma | is_localization.integer_normalization_aeval_eq_zero | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"is_scalar_tower",
"is_scalar_tower.algebra_map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integer_normalization_eq_zero_iff {p : K[X]} :
integer_normalization (non_zero_divisors A) p = 0 ↔ p = 0 | begin
refine (polynomial.ext_iff.trans (polynomial.ext_iff.trans _).symm),
obtain ⟨⟨b, nonzero⟩, hb⟩ := integer_normalization_spec _ p,
split; intros h i,
{ apply to_map_eq_zero_iff.mp,
rw [hb i, h i],
apply smul_zero,
assumption },
{ have hi := h i,
rw [polynomial.coeff_zero, ← @to_map_eq_zer... | lemma | is_fraction_ring.integer_normalization_eq_zero_iff | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra.smul_def",
"non_zero_divisors",
"polynomial.coeff_zero",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_iff [algebra A C] [algebra K C] [is_scalar_tower A K C] {x : C} :
is_algebraic A x ↔ is_algebraic K x | begin
split; rintros ⟨p, hp, px⟩,
{ refine ⟨p.map (algebra_map A K), λ h, hp (polynomial.ext (λ i, _)), _⟩,
{ have : algebra_map A K (p.coeff i) = 0 := trans (polynomial.coeff_map _ _).symm (by simp [h]),
exact to_map_eq_zero_iff.mp this },
{ exact (polynomial.aeval_map_algebra_map K _ _).trans px, } ... | lemma | is_fraction_ring.is_algebraic_iff | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"algebra_map",
"is_algebraic",
"is_scalar_tower",
"polynomial.aeval_map_algebra_map",
"polynomial.coeff_map",
"polynomial.ext"
] | An element of a ring is algebraic over the ring `A` iff it is algebraic
over the field of fractions of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_is_algebraic_iff [algebra A C] [algebra K C] [is_scalar_tower A K C] :
algebra.is_algebraic A C ↔ algebra.is_algebraic K C | ⟨λ h x, (is_algebraic_iff A K C).mp (h x), λ h x, (is_algebraic_iff A K C).mpr (h x)⟩ | lemma | is_fraction_ring.comap_is_algebraic_iff | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"algebra.is_algebraic",
"is_scalar_tower"
] | A ring is algebraic over the ring `A` iff it is algebraic over the field of fractions of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.is_integral_elem_localization_at_leading_coeff
{R S : Type*} [comm_ring R] [comm_ring S] (f : R →+* S)
(x : S) (p : R[X]) (hf : p.eval₂ f x = 0) (M : submonoid R)
(hM : p.leading_coeff ∈ M) {Rₘ Sₘ : Type*} [comm_ring Rₘ] [comm_ring Sₘ]
[algebra R Rₘ] [is_localization M Rₘ]
[algebra S Sₘ] [is_localiza... | begin
by_cases triv : (1 : Rₘ) = 0,
{ exact ⟨0, ⟨trans leading_coeff_zero triv.symm, eval₂_zero _ _⟩⟩ },
haveI : nontrivial Rₘ := nontrivial_of_ne 1 0 triv,
obtain ⟨b, hb⟩ := is_unit_iff_exists_inv.mp
(map_units Rₘ ⟨p.leading_coeff, hM⟩),
refine ⟨(p.map (algebra_map R Rₘ)) * C b, ⟨_, _⟩⟩,
{ refine monic... | lemma | ring_hom.is_integral_elem_localization_at_leading_coeff | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"algebra_map",
"comm_ring",
"is_localization",
"is_localization.map_comp",
"nontrivial",
"nontrivial_of_ne",
"ring_hom.map_zero",
"submonoid",
"zero_mul",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_localization_at_leading_coeff {x : S} (p : R[X])
(hp : aeval x p = 0) (hM : p.leading_coeff ∈ M) :
(map Sₘ (algebra_map R S)
(show _ ≤ (algebra.algebra_map_submonoid S M).comap _, from M.le_comap_map)
: Rₘ →+* _).is_integral_elem (algebra_map S Sₘ x) | (algebra_map R S).is_integral_elem_localization_at_leading_coeff x p hp M hM | theorem | is_integral_localization_at_leading_coeff | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra.algebra_map_submonoid",
"algebra_map"
] | Given a particular witness to an element being algebraic over an algebra `R → S`,
We can localize to a submonoid containing the leading coefficient to make it integral.
Explicitly, the map between the localizations will be an integral ring morphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_localization (H : algebra.is_integral R S) :
(map Sₘ (algebra_map R S)
(show _ ≤ (algebra.algebra_map_submonoid S M).comap _, from M.le_comap_map)
: Rₘ →+* _).is_integral | begin
intro x,
obtain ⟨⟨s, ⟨u, hu⟩⟩, hx⟩ := surj (algebra.algebra_map_submonoid S M) x,
obtain ⟨v, hv⟩ := hu,
obtain ⟨v', hv'⟩ := is_unit_iff_exists_inv'.1 (map_units Rₘ ⟨v, hv.1⟩),
refine @is_integral_of_is_integral_mul_unit Rₘ _ _ _
(localization_algebra M S) x (algebra_map S Sₘ u) v' _ _,
{ replace h... | theorem | is_integral_localization | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra.algebra_map_submonoid",
"algebra.is_integral",
"algebra_map",
"is_integral",
"is_integral_localization_at_leading_coeff",
"is_integral_of_is_integral_mul_unit",
"is_localization.map_comp",
"localization_algebra",
"ring_hom.comp_apply",
"ring_hom.map_mul",
"ring_hom.map_one"
] | If `R → S` is an integral extension, `M` is a submonoid of `R`,
`Rₘ` is the localization of `R` at `M`,
and `Sₘ` is the localization of `S` at the image of `M` under the extension map,
then the induced map `Rₘ → Sₘ` is also an integral extension | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_localization' {R S : Type*} [comm_ring R] [comm_ring S]
{f : R →+* S} (hf : f.is_integral) (M : submonoid R) :
(map (localization (M.map (f : R →* S))) f
(M.le_comap_map : _ ≤ submonoid.comap (f : R →* S) _) : localization M →+* _).is_integral | @is_integral_localization R _ M S _ f.to_algebra _ _ _ _ _ _ _ _ hf | lemma | is_integral_localization' | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"comm_ring",
"is_integral",
"is_integral_localization",
"localization",
"submonoid",
"submonoid.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.scale_roots_common_denom_mem_lifts (p : Rₘ[X])
(hp : p.leading_coeff ∈ (algebra_map R Rₘ).range) :
p.scale_roots (algebra_map R Rₘ $ is_localization.common_denom M p.support p.coeff) ∈
polynomial.lifts (algebra_map R Rₘ) | begin
rw polynomial.lifts_iff_coeff_lifts,
intro n,
rw [polynomial.coeff_scale_roots],
by_cases h₁ : n ∈ p.support,
by_cases h₂ : n = p.nat_degree,
{ rwa [h₂, polynomial.coeff_nat_degree, tsub_self, pow_zero, _root_.mul_one] },
{ have : n + 1 ≤ p.nat_degree := lt_of_le_of_ne (polynomial.le_nat_degree_of_m... | lemma | is_localization.scale_roots_common_denom_mem_lifts | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra.smul_def",
"algebra_map",
"is_localization.common_denom",
"is_localization.map_integer_multiple",
"le_tsub_of_add_le_left",
"map_pow",
"mul_comm",
"polynomial.coeff_nat_degree",
"polynomial.coeff_scale_roots",
"polynomial.le_nat_degree_of_mem_supp",
"polynomial.lifts",
"polynomial.lif... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.exists_multiple_integral_of_is_localization
[algebra Rₘ S] [is_scalar_tower R Rₘ S] (x : S) (hx : is_integral Rₘ x) :
∃ m : M, is_integral R (m • x) | begin
cases subsingleton_or_nontrivial Rₘ with _ nontriv; resetI,
{ haveI := (algebra_map Rₘ S).codomain_trivial,
exact ⟨1, polynomial.X, polynomial.monic_X, subsingleton.elim _ _⟩ },
obtain ⟨p, hp₁, hp₂⟩ := hx,
obtain ⟨p', hp'₁, -, hp'₂⟩ := lifts_and_nat_degree_eq_and_monic
(is_localization.scale_roots... | lemma | is_integral.exists_multiple_integral_of_is_localization | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"is_integral",
"is_localization.scale_roots_common_denom_mem_lifts",
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply",
"is_scalar_tower.algebra_map_eq",
"polynomial.X",
"polynomial.eval₂_map",
"polynomial.monic_X",
"polynomial.monic_scale_roo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fraction_ring_of_algebraic (alg : is_algebraic A L)
(inj : ∀ x, algebra_map A L x = 0 → x = 0) :
is_fraction_ring C L | { map_units := λ ⟨y, hy⟩,
is_unit.mk0 _ (show algebra_map C L y ≠ 0, from λ h, mem_non_zero_divisors_iff_ne_zero.mp hy
((injective_iff_map_eq_zero (algebra_map C L)).mp (algebra_map_injective C A L) _ h)),
surj := λ z, let ⟨x, y, hy, hxy⟩ := exists_integral_multiple (alg z) inj in
⟨⟨mk' C (x : L) x.2, a... | lemma | is_integral_closure.is_fraction_ring_of_algebraic | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"algebra_map_injective",
"exists_integral_multiple",
"is_algebraic",
"is_fraction_ring",
"is_scalar_tower.algebra_map_apply",
"is_unit.mk0",
"mul_left_cancel₀",
"ring_hom.map_zero",
"set_like.coe_mk"
] | If the field `L` is an algebraic extension of the integral domain `A`,
the integral closure `C` of `A` in `L` has fraction field `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fraction_ring_of_finite_extension [algebra K L] [is_scalar_tower A K L]
[finite_dimensional K L] : is_fraction_ring C L | is_fraction_ring_of_algebraic A C
(is_fraction_ring.comap_is_algebraic_iff.mpr (is_algebraic_of_finite K L))
(λ x hx, is_fraction_ring.to_map_eq_zero_iff.mp ((map_eq_zero $ algebra_map K L).mp $
(is_scalar_tower.algebra_map_apply _ _ _ _).symm.trans hx)) | lemma | is_integral_closure.is_fraction_ring_of_finite_extension | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"algebra_map",
"finite_dimensional",
"is_fraction_ring",
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply",
"map_eq_zero"
] | If the field `L` is a finite extension of the fraction field of the integral domain `A`,
the integral closure `C` of `A` in `L` has fraction field `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fraction_ring_of_algebraic [algebra A L] (alg : is_algebraic A L)
(inj : ∀ x, algebra_map A L x = 0 → x = 0) :
is_fraction_ring (integral_closure A L) L | is_integral_closure.is_fraction_ring_of_algebraic A (integral_closure A L) alg inj | lemma | integral_closure.is_fraction_ring_of_algebraic | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"algebra_map",
"integral_closure",
"is_algebraic",
"is_fraction_ring",
"is_integral_closure.is_fraction_ring_of_algebraic"
] | If the field `L` is an algebraic extension of the integral domain `A`,
the integral closure of `A` in `L` has fraction field `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fraction_ring_of_finite_extension [algebra A L] [algebra K L]
[is_scalar_tower A K L] [finite_dimensional K L] :
is_fraction_ring (integral_closure A L) L | is_integral_closure.is_fraction_ring_of_finite_extension A K L (integral_closure A L) | lemma | integral_closure.is_fraction_ring_of_finite_extension | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"finite_dimensional",
"integral_closure",
"is_fraction_ring",
"is_integral_closure.is_fraction_ring_of_finite_extension",
"is_scalar_tower"
] | If the field `L` is a finite extension of the fraction field of the integral domain `A`,
the integral closure of `A` in `L` has fraction field `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_algebraic_iff' [field K] [is_domain R] [is_domain S] [algebra R K] [algebra S K]
[no_zero_smul_divisors R K] [is_fraction_ring S K] [is_scalar_tower R S K] :
algebra.is_algebraic R S ↔ algebra.is_algebraic R K | begin
simp only [algebra.is_algebraic],
split,
{ intros h x,
rw [is_fraction_ring.is_algebraic_iff R (fraction_ring R) K, is_algebraic_iff_is_integral],
obtain ⟨(a : S), b, ha, rfl⟩ := @div_surjective S _ _ _ _ _ _ x,
obtain ⟨f, hf₁, hf₂⟩ := h b,
rw [div_eq_mul_inv],
refine is_integral_mul _ _... | lemma | is_fraction_ring.is_algebraic_iff' | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"algebra.is_algebraic",
"algebra_map",
"div_eq_mul_inv",
"field",
"fraction_ring",
"inv_of_eq_inv",
"invertible",
"is_algebraic_algebra_map_of_is_algebraic",
"is_algebraic_iff_is_integral",
"is_domain",
"is_fraction_ring",
"is_fraction_ring.is_algebraic_iff",
"is_integral_mul",
... | `S` is algebraic over `R` iff a fraction ring of `S` is algebraic over `R` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_span_singleton_map_subset {L : Type*}
[is_domain R] [is_domain S] [field K] [field L]
[algebra R K] [algebra R L] [algebra S L] [is_integral_closure S R L]
[is_fraction_ring S L] [algebra K L] [is_scalar_tower R S L] [is_scalar_tower R K L]
{a : S} {b : set S} (alg : algebra.is_algebraic R L) (inj : funct... | begin
intros x hx,
obtain ⟨x', rfl⟩ := ideal.mem_span_singleton.mp hx,
obtain ⟨y', z', rfl⟩ := is_localization.mk'_surjective (S⁰) x',
obtain ⟨y, z, hz0, yz_eq⟩ := is_integral_closure.exists_smul_eq_mul alg inj y'
(non_zero_divisors.coe_ne_zero z'),
have injRS : function.injective (algebra_map R S),
{ r... | lemma | is_fraction_ring.ideal_span_singleton_map_subset | ring_theory.localization | src/ring_theory/localization/integral.lean | [
"data.polynomial.lifts",
"group_theory.monoid_localization",
"ring_theory.algebraic",
"ring_theory.ideal.local_ring",
"ring_theory.integral_closure",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra",
"algebra.is_algebraic",
"algebra.smul_def",
"algebra_map",
"div_eq_mul_inv",
"field",
"function.injective.of_comp",
"ideal.span",
"is_domain",
"is_fraction_ring",
"is_fraction_ring.mk'_eq_div",
"is_integral_closure",
"is_integral_closure.exists_smul_eq_mul",
"is_localization.mk'... | If the `S`-multiples of `a` are contained in some `R`-span, then `Frac(S)`-multiples of `a`
are contained in the equivalent `Frac(R)`-span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_submonoid : submonoid S | (M.map (algebra_map R S)).left_inv | def | is_localization.inv_submonoid | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"algebra_map",
"submonoid"
] | The submonoid of `S = M⁻¹R` consisting of `{ 1 / x | x ∈ M }`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submonoid_map_le_is_unit : M.map (algebra_map R S) ≤ is_unit.submonoid S | by { rintros _ ⟨a, ha, rfl⟩, exact is_localization.map_units S ⟨_, ha⟩ } | lemma | is_localization.submonoid_map_le_is_unit | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"algebra_map",
"is_unit.submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_inv_submonoid : M.map (algebra_map R S) ≃* inv_submonoid M S | ((M.map (algebra_map R S)).left_inv_equiv (submonoid_map_le_is_unit M S)).symm | abbreviation | is_localization.equiv_inv_submonoid | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"algebra_map"
] | There is an equivalence of monoids between the image of `M` and `inv_submonoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_inv_submonoid : M →* inv_submonoid M S | (equiv_inv_submonoid M S).to_monoid_hom.comp ((algebra_map R S : R →* S).submonoid_map M) | def | is_localization.to_inv_submonoid | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"algebra_map"
] | There is a canonical map from `M` to `inv_submonoid` sending `x` to `1 / x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_inv_submonoid_surjective : function.surjective (to_inv_submonoid M S) | function.surjective.comp (equiv.surjective _) (monoid_hom.submonoid_map_surjective _ _) | lemma | is_localization.to_inv_submonoid_surjective | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"equiv.surjective",
"monoid_hom.submonoid_map_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_inv_submonoid_mul (m : M) : (to_inv_submonoid M S m : S) * (algebra_map R S m) = 1 | submonoid.left_inv_equiv_symm_mul _ _ _ | lemma | is_localization.to_inv_submonoid_mul | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"algebra_map",
"submonoid.left_inv_equiv_symm_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_to_inv_submonoid (m : M) : (algebra_map R S m) * (to_inv_submonoid M S m : S) = 1 | submonoid.mul_left_inv_equiv_symm _ _ ⟨_, _⟩ | lemma | is_localization.mul_to_inv_submonoid | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"algebra_map",
"submonoid.mul_left_inv_equiv_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_to_inv_submonoid (m : M) : m • (to_inv_submonoid M S m : S) = 1 | by { convert mul_to_inv_submonoid M S m, rw ← algebra.smul_def, refl } | lemma | is_localization.smul_to_inv_submonoid | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"algebra.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surj' (z : S) : ∃ (r : R) (m : M), z = r • to_inv_submonoid M S m | begin
rcases is_localization.surj M z with ⟨⟨r, m⟩, e : z * _ = algebra_map R S r⟩,
refine ⟨r, m, _⟩,
rw [algebra.smul_def, ← e, mul_assoc],
simp,
end | lemma | is_localization.surj' | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"algebra.smul_def",
"algebra_map",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_inv_submonoid_eq_mk' (x : M) :
(to_inv_submonoid M S x : S) = mk' S 1 x | by { rw ← (is_localization.map_units S x).mul_left_inj, simp } | lemma | is_localization.to_inv_submonoid_eq_mk' | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"mk'",
"mul_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inv_submonoid_iff_exists_mk' (x : S) :
x ∈ inv_submonoid M S ↔ ∃ m : M, mk' S 1 m = x | begin
simp_rw ← to_inv_submonoid_eq_mk',
exact ⟨λ h, ⟨_, congr_arg subtype.val (to_inv_submonoid_surjective M S ⟨x, h⟩).some_spec⟩,
λ h, h.some_spec ▸ (to_inv_submonoid M S h.some).prop⟩
end | lemma | is_localization.mem_inv_submonoid_iff_exists_mk' | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_inv_submonoid : submodule.span R (inv_submonoid M S : set S) = ⊤ | begin
rw eq_top_iff,
rintros x -,
rcases is_localization.surj' M x with ⟨r, m, rfl⟩,
exact submodule.smul_mem _ _ (submodule.subset_span (to_inv_submonoid M S m).prop),
end | lemma | is_localization.span_inv_submonoid | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"eq_top_iff",
"is_localization.surj'",
"submodule.smul_mem",
"submodule.span",
"submodule.subset_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_type_of_monoid_fg [monoid.fg M] : algebra.finite_type R S | begin
have := monoid.fg_of_surjective _ (to_inv_submonoid_surjective M S),
rw monoid.fg_iff_submonoid_fg at this,
rcases this with ⟨s, hs⟩,
refine ⟨⟨s, _⟩⟩,
rw eq_top_iff,
rintro x -,
change x ∈ ((algebra.adjoin R _ : subalgebra R S).to_submodule : set S),
rw [algebra.adjoin_eq_span, hs, span_inv_submon... | lemma | is_localization.finite_type_of_monoid_fg | ring_theory.localization | src/ring_theory/localization/inv_submonoid.lean | [
"group_theory.submonoid.inverses",
"ring_theory.finite_type",
"ring_theory.localization.basic",
"tactic.ring_exp"
] | [
"algebra.adjoin",
"algebra.adjoin_eq_span",
"algebra.finite_type",
"eq_top_iff",
"monoid.fg",
"monoid.fg_iff_submonoid_fg",
"monoid.fg_of_surjective",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_localization_submodule : submonoid R | (N ⊔ M.map (algebra_map R S)).comap (algebra_map R S) | def | is_localization.localization_localization_submodule | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_localization_localization_submodule {x : R} :
x ∈ localization_localization_submodule M N ↔
∃ (y : N) (z : M), algebra_map R S x = y * algebra_map R S z | begin
rw [localization_localization_submodule, submonoid.mem_comap, submonoid.mem_sup],
split,
{ rintros ⟨y, hy, _, ⟨z, hz, rfl⟩, e⟩, exact ⟨⟨y, hy⟩, ⟨z, hz⟩ ,e.symm⟩ },
{ rintros ⟨y, z, e⟩, exact ⟨y, y.prop, _, ⟨z, z.prop, rfl⟩, e.symm⟩ }
end | lemma | is_localization.mem_localization_localization_submodule | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"submonoid.mem_comap",
"submonoid.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_localization_map_units [is_localization N T]
(y : localization_localization_submodule M N) : is_unit (algebra_map R T y) | begin
obtain ⟨y', z, eq⟩ := mem_localization_localization_submodule.mp y.prop,
rw [is_scalar_tower.algebra_map_apply R S T, eq, ring_hom.map_mul, is_unit.mul_iff],
exact ⟨is_localization.map_units T y',
(is_localization.map_units _ z).map (algebra_map S T)⟩,
end | lemma | is_localization.localization_localization_map_units | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"is_localization",
"is_scalar_tower.algebra_map_apply",
"is_unit",
"is_unit.mul_iff",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_localization_surj [is_localization N T] (x : T) :
∃ (y : R × localization_localization_submodule M N),
x * (algebra_map R T y.2) = algebra_map R T y.1 | begin
rcases is_localization.surj N x with ⟨⟨y, s⟩, eq₁⟩, -- x = y / s
rcases is_localization.surj M y with ⟨⟨z, t⟩, eq₂⟩, -- y = z / t
rcases is_localization.surj M (s : S) with ⟨⟨z', t'⟩, eq₃⟩, -- s = z' / t'
dsimp only at eq₁ eq₂ eq₃,
use z * t', use z' * t, -- x = y / s = (z * t') / (z' * t)
{ rw mem_lo... | lemma | is_localization.localization_localization_surj | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"is_localization",
"is_scalar_tower.algebra_map_apply",
"mul_assoc",
"mul_comm",
"ring",
"ring_hom.map_mul",
"submonoid.coe_mul",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_localization_eq_iff_exists [is_localization N T] (x y : R) :
algebra_map R T x = algebra_map R T y ↔
∃ (c : localization_localization_submodule M N), ↑c * x = ↑c * y | begin
rw [is_scalar_tower.algebra_map_apply R S T, is_scalar_tower.algebra_map_apply R S T,
is_localization.eq_iff_exists N T],
split,
{ rintros ⟨z, eq₁⟩,
rcases is_localization.surj M (z : S) with ⟨⟨z', s⟩, eq₂⟩,
dsimp only at eq₂,
obtain ⟨c, eq₃ : ↑c * (x * z') = ↑c * (y * z')⟩ := (is_localiz... | lemma | is_localization.localization_localization_eq_iff_exists | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"is_localization",
"is_scalar_tower.algebra_map_apply",
"map_mul",
"mul_comm",
"mul_left_comm",
"submonoid.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_localization_is_localization [is_localization N T] :
is_localization (localization_localization_submodule M N) T | { map_units := localization_localization_map_units M N T,
surj := localization_localization_surj M N T,
eq_iff_exists := localization_localization_eq_iff_exists M N T } | lemma | is_localization.localization_localization_is_localization | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"is_localization"
] | Given submodules `M ⊆ R` and `N ⊆ S = M⁻¹R`, with `f : R →+* S` the localization map, we have
`N ⁻¹ S = T = (f⁻¹ (N • f(M))) ⁻¹ R`. I.e., the localization of a localization is a localization. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization_localization_is_localization_of_has_all_units
[is_localization N T] (H : ∀ (x : S), is_unit x → x ∈ N) :
is_localization (N.comap (algebra_map R S)) T | begin
convert localization_localization_is_localization M N T,
symmetry,
rw sup_eq_left,
rintros _ ⟨x, hx, rfl⟩,
exact H _ (is_localization.map_units _ ⟨x, hx⟩),
end | lemma | is_localization.localization_localization_is_localization_of_has_all_units | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"is_localization",
"is_unit",
"sup_eq_left"
] | Given submodules `M ⊆ R` and `N ⊆ S = M⁻¹R`, with `f : R →+* S` the localization map, if
`N` contains all the units of `S`, then `N ⁻¹ S = T = (f⁻¹ N) ⁻¹ R`. I.e., the localization of a
localization is a localization. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization_is_localization_at_prime_is_localization (p : ideal S) [Hp : p.is_prime]
[is_localization.at_prime T p] :
is_localization.at_prime T (p.comap (algebra_map R S)) | begin
apply localization_localization_is_localization_of_has_all_units M p.prime_compl T,
intros x hx hx',
exact (Hp.1 : ¬ _) (p.eq_top_of_is_unit_mem hx' hx),
end | lemma | is_localization.is_localization_is_localization_at_prime_is_localization | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"ideal",
"is_localization.at_prime"
] | Given a submodule `M ⊆ R` and a prime ideal `p` of `S = M⁻¹R`, with `f : R →+* S` the localization
map, then `T = Sₚ` is the localization of `R` at `f⁻¹(p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization_localization_at_prime_is_localization (p : ideal (localization M))
[p.is_prime] : is_localization.at_prime (localization.at_prime p) (p.comap (algebra_map R _)) | is_localization_is_localization_at_prime_is_localization M _ _ | instance | is_localization.localization_localization_at_prime_is_localization | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"ideal",
"is_localization.at_prime",
"localization",
"localization.at_prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_localization_at_prime_iso_localization (p : ideal (localization M)) [p.is_prime] :
localization.at_prime (p.comap (algebra_map R (localization M))) ≃ₐ[R] localization.at_prime p | is_localization.alg_equiv (p.comap (algebra_map R (localization M))).prime_compl _ _ | def | is_localization.localization_localization_at_prime_iso_localization | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"ideal",
"is_localization.alg_equiv",
"localization",
"localization.at_prime"
] | Given a submodule `M ⊆ R` and a prime ideal `p` of `M⁻¹R`, with `f : R →+* S` the localization
map, then `(M⁻¹R)ₚ` is isomorphic (as an `R`-algebra) to the localization of `R` at `f⁻¹(p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization_algebra_of_submonoid_le
(M N : submonoid R) (h : M ≤ N) [is_localization M S] [is_localization N T] :
algebra S T | (is_localization.lift (λ y, (map_units T ⟨↑y, h y.prop⟩ : _)) : S →+* T).to_algebra | def | is_localization.localization_algebra_of_submonoid_le | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra",
"is_localization",
"is_localization.lift",
"submonoid"
] | Given submonoids `M ≤ N` of `R`, this is the canonical algebra structure
of `M⁻¹S` acting on `N⁻¹S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization_is_scalar_tower_of_submonoid_le
(M N : submonoid R) (h : M ≤ N) [is_localization M S] [is_localization N T] :
@@is_scalar_tower R S T _ (localization_algebra_of_submonoid_le S T M N h).to_has_smul _ | begin
letI := localization_algebra_of_submonoid_le S T M N h,
exact is_scalar_tower.of_algebra_map_eq' (is_localization.lift_comp _).symm
end | lemma | is_localization.localization_is_scalar_tower_of_submonoid_le | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"is_localization",
"is_localization.lift_comp",
"is_scalar_tower",
"is_scalar_tower.of_algebra_map_eq'",
"submonoid"
] | If `M ≤ N` are submonoids of `R`, then the natural map `M⁻¹S →+* N⁻¹S` commutes with the
localization maps | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization_of_submonoid_le
(M N : submonoid R) (h : M ≤ N) [is_localization M S] [is_localization N T]
[algebra S T] [is_scalar_tower R S T] :
is_localization (N.map (algebra_map R S)) T | { map_units := begin
rintro ⟨_, ⟨y, hy, rfl⟩⟩,
convert is_localization.map_units T ⟨y, hy⟩,
exact (is_scalar_tower.algebra_map_apply _ _ _ _).symm
end,
surj := λ y, begin
obtain ⟨⟨x, s⟩, e⟩ := is_localization.surj N y,
refine ⟨⟨algebra_map _ _ x, _, _, s.prop, rfl⟩, _⟩,
simpa [← is_scalar_to... | lemma | is_localization.is_localization_of_submonoid_le | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra",
"algebra_map",
"is_localization",
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply",
"map_mul",
"mul_assoc",
"mul_comm",
"mul_left_inj",
"mul_right_comm",
"ring",
"set.exists_image_iff",
"submonoid",
"subtype.coe_mk"
] | If `M ≤ N` are submonoids of `R`, then `N⁻¹S` is also the localization of `M⁻¹S` at `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization_of_is_exists_mul_mem (M N : submonoid R) [is_localization M S] (h : M ≤ N)
(h' : ∀ x : N, ∃ m : R, m * x ∈ M) : is_localization N S | { map_units := λ y, begin
obtain ⟨m, hm⟩ := h' y,
have := is_localization.map_units S ⟨_, hm⟩,
erw map_mul at this,
exact (is_unit.mul_iff.mp this).2
end,
surj := λ z, by { obtain ⟨⟨y, s⟩, e⟩ := is_localization.surj M z, exact ⟨⟨y, _, h s.prop⟩, e⟩ },
eq_iff_exists := λ x₁ x₂, begin
rw is_loca... | lemma | is_localization.is_localization_of_is_exists_mul_mem | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"is_localization",
"map_mul",
"mul_assoc",
"submonoid"
] | If `M ≤ N` are submonoids of `R` such that `∀ x : N, ∃ m : R, m * x ∈ M`, then the
localization at `N` is equal to the localizaton of `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fraction_ring_of_is_localization (S T : Type*) [comm_ring S] [comm_ring T]
[algebra R S] [algebra R T] [algebra S T] [is_scalar_tower R S T]
[is_localization M S] [is_fraction_ring R T] (hM : M ≤ non_zero_divisors R) :
is_fraction_ring S T | begin
have := is_localization_of_submonoid_le S T M (non_zero_divisors R) _,
refine @@is_localization_of_is_exists_mul_mem _ _ _ _ _ _ this _ _,
{ exact map_non_zero_divisors_le M S },
{ rintro ⟨x, hx⟩,
obtain ⟨⟨y, s⟩, e⟩ := is_localization.surj M x,
use algebra_map R S s,
rw [mul_comm, subtype.coe_... | lemma | is_fraction_ring.is_fraction_ring_of_is_localization | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra",
"algebra_map",
"comm_ring",
"is_fraction_ring",
"is_localization",
"is_localization.injective",
"is_scalar_tower",
"map_mul",
"mul_assoc",
"mul_comm",
"mul_left_inj",
"non_zero_divisors",
"set.mem_image_of_mem",
"subtype.coe_mk",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fraction_ring_of_is_domain_of_is_localization [is_domain R] (S T : Type*)
[comm_ring S] [comm_ring T] [algebra R S] [algebra R T] [algebra S T]
[is_scalar_tower R S T] [is_localization M S] [is_fraction_ring R T] : is_fraction_ring S T | begin
haveI := is_fraction_ring.nontrivial R T,
haveI := (algebra_map S T).domain_nontrivial,
apply is_fraction_ring_of_is_localization M S T,
intros x hx,
rw mem_non_zero_divisors_iff_ne_zero,
intro hx',
apply @zero_ne_one S,
rw [← (algebra_map R S).map_one, ← @mk'_one R _ M, @comm _ eq, mk'_eq_zero_if... | lemma | is_fraction_ring.is_fraction_ring_of_is_domain_of_is_localization | ring_theory.localization | src/ring_theory/localization/localization_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.localization.basic",
"ring_theory.localization.fraction_ring"
] | [
"algebra",
"algebra_map",
"comm",
"comm_ring",
"is_domain",
"is_fraction_ring",
"is_fraction_ring.nontrivial",
"is_localization",
"is_scalar_tower",
"map_one",
"mem_non_zero_divisors_iff_ne_zero",
"mk'_one",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent.localization {ι : Type*} {b : ι → M} (hli : linear_independent R b) :
linear_independent Rₛ b | begin
rw linear_independent_iff' at ⊢ hli,
intros s g hg i hi,
choose! a g' hg' using is_localization.exist_integer_multiples S s g,
specialize hli s g' _ i hi,
{ rw [← @smul_zero _ M _ _ (a : R), ← hg, finset.smul_sum],
refine finset.sum_congr rfl (λ i hi, _),
rw [← is_scalar_tower.algebra_map_smul R... | lemma | linear_independent.localization | ring_theory.localization | src/ring_theory/localization/module.lean | [
"linear_algebra.basis",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer"
] | [
"algebra.smul_def",
"algebra_map",
"finset.smul_sum",
"is_localization.exist_integer_multiples",
"is_scalar_tower.algebra_map_smul",
"linear_independent",
"linear_independent_iff'",
"smul_assoc",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent.localization_localization
{ι : Type*} {v : ι → A} (hv : linear_independent R v) :
linear_independent Rₛ (algebra_map A Aₛ ∘ v) | begin
rw linear_independent_iff' at ⊢ hv,
intros s g hg i hi,
choose! a g' hg' using is_localization.exist_integer_multiples S s g,
have h0 : algebra_map A Aₛ (∑ i in s, g' i • v i) = 0,
{ apply_fun ((•) (a : R)) at hg,
rw [smul_zero, finset.smul_sum] at hg,
rw [map_sum, ← hg],
refine finset.sum_c... | lemma | linear_independent.localization_localization | ring_theory.localization | src/ring_theory/localization/module.lean | [
"linear_algebra.basis",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer"
] | [
"algebra.algebra_map_submonoid",
"algebra.smul_def",
"algebra_map",
"algebra_map_smul",
"finset.mul_sum",
"finset.smul_sum",
"is_localization.exist_integer_multiples",
"is_localization.map_eq_zero_iff",
"is_scalar_tower.algebra_map_apply",
"linear_independent",
"linear_independent_iff'",
"map_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_eq_top.localization_localization {v : set A} (hv : span R v = ⊤) :
span Rₛ (algebra_map A Aₛ '' v) = ⊤ | begin
rw eq_top_iff,
rintros a' -,
obtain ⟨a, ⟨_, s, hs, rfl⟩, rfl⟩ := is_localization.mk'_surjective
(algebra.algebra_map_submonoid A S) a',
rw [is_localization.mk'_eq_mul_mk'_one, mul_comm, ← map_one (algebra_map R A)],
erw ← is_localization.algebra_map_mk' A Rₛ Aₛ (1 : R) ⟨s, hs⟩, -- `erw` needed to un... | lemma | span_eq_top.localization_localization | ring_theory.localization | src/ring_theory/localization/module.lean | [
"linear_algebra.basis",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer"
] | [
"algebra.algebra_map_submonoid",
"algebra.coe_linear_map",
"algebra.smul_def",
"algebra_map",
"eq_top_iff",
"is_localization.algebra_map_mk'",
"is_localization.mk'_eq_mul_mk'_one",
"is_localization.mk'_surjective",
"linear_map.coe_restrict_scalars",
"map_one",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.localization_localization {ι : Type*} (b : basis ι R A) : basis ι Rₛ Aₛ | basis.mk
(b.linear_independent.localization_localization _ S _)
(by { rw [set.range_comp, span_eq_top.localization_localization Rₛ S Aₛ b.span_eq],
exact le_rfl }) | def | basis.localization_localization | ring_theory.localization | src/ring_theory/localization/module.lean | [
"linear_algebra.basis",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer"
] | [
"basis",
"basis.mk",
"le_rfl",
"set.range_comp",
"span_eq_top.localization_localization"
] | If `A` has an `R`-basis, then localizing `A` at `S` has a basis over `R` localized at `S`.
A suitable instance for `[algebra A Aₛ]` is `localization_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis.localization_localization_apply {ι : Type*} (b : basis ι R A) (i) :
b.localization_localization Rₛ S Aₛ i = algebra_map A Aₛ (b i) | basis.mk_apply _ _ _ | lemma | basis.localization_localization_apply | ring_theory.localization | src/ring_theory/localization/module.lean | [
"linear_algebra.basis",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer"
] | [
"algebra_map",
"basis",
"basis.mk_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis.localization_localization_repr_algebra_map
{ι : Type*} (b : basis ι R A) (x i) :
(b.localization_localization Rₛ S Aₛ).repr (algebra_map A Aₛ x) i =
algebra_map R Rₛ (b.repr x i) | calc (b.localization_localization Rₛ S Aₛ).repr (algebra_map A Aₛ x) i
= (b.localization_localization Rₛ S Aₛ).repr
((b.repr x).sum (λ j c, algebra_map R Rₛ c • algebra_map A Aₛ (b j))) i :
by simp_rw [is_scalar_tower.algebra_map_smul, algebra.smul_def,
is_scalar_tower.algebra_map_apply R A ... | lemma | basis.localization_localization_repr_algebra_map | ring_theory.localization | src/ring_theory/localization/module.lean | [
"linear_algebra.basis",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer"
] | [
"algebra.smul_def",
"algebra_map",
"basis",
"basis.repr_self",
"basis.total_repr",
"finsupp.single",
"finsupp.smul_apply",
"finsupp.sum_apply",
"finsupp.total_apply",
"is_scalar_tower.algebra_map_apply",
"is_scalar_tower.algebra_map_smul",
"linear_equiv.map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent.iff_fraction_ring {ι : Type*} {b : ι → V} :
linear_independent R b ↔ linear_independent K b | ⟨linear_independent.localization K (R⁰),
linear_independent.restrict_scalars (smul_left_injective R one_ne_zero)⟩ | lemma | linear_independent.iff_fraction_ring | ring_theory.localization | src/ring_theory/localization/module.lean | [
"linear_algebra.basis",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer"
] | [
"linear_independent",
"linear_independent.restrict_scalars",
"one_ne_zero",
"smul_left_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.norm_localization [module.free R S] [module.finite R S] (a : S) :
algebra.norm Rₘ (algebra_map S Sₘ a) = algebra_map R Rₘ (algebra.norm R a) | begin
casesI subsingleton_or_nontrivial R,
{ haveI : subsingleton Rₘ := module.subsingleton R Rₘ,
simp },
let b := module.free.choose_basis R S,
letI := classical.dec_eq (module.free.choose_basis_index R S),
rw [algebra.norm_eq_matrix_det (b.localization_localization Rₘ M Sₘ),
algebra.norm_eq_matrix... | lemma | algebra.norm_localization | ring_theory.localization | src/ring_theory/localization/norm.lean | [
"ring_theory.localization.module",
"ring_theory.norm"
] | [
"algebra.left_mul_matrix_eq_repr_mul",
"algebra.norm",
"algebra.norm_eq_matrix_det",
"algebra_map",
"basis.localization_localization_apply",
"basis.localization_localization_repr_algebra_map",
"classical.dec_eq",
"matrix.map_apply",
"module.finite",
"module.free",
"module.free.choose_basis",
"... | Let `S` be an extension of `R` and `Rₘ Sₘ` be localizations at `M` of `R S` respectively.
Then the norm of `a : Sₘ` over `Rₘ` is the norm of `a : S` over `R` if `S` is free as `R`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_reduced_fraction (x : K) :
∃ (a : A) (b : non_zero_divisors A),
(∀ {d}, d ∣ a → d ∣ b → is_unit d) ∧ mk' K a b = x | begin
obtain ⟨⟨b, b_nonzero⟩, a, hab⟩ := exists_integer_multiple (non_zero_divisors A) x,
obtain ⟨a', b', c', no_factor, rfl, rfl⟩ :=
unique_factorization_monoid.exists_reduced_factors' a b
(mem_non_zero_divisors_iff_ne_zero.mp b_nonzero),
obtain ⟨c'_nonzero, b'_nonzero⟩ := mul_mem_non_zero_divisors.mp ... | lemma | is_fraction_ring.exists_reduced_fraction | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"algebra.smul_def",
"is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors",
"is_unit",
"mk'",
"mul_assoc",
"mul_left_cancel₀",
"non_zero_divisors",
"ring_hom.map_mul",
"subtype.coe_mk",
"unique_factorization_monoid.exists_reduced_factors'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num (x : K) : A | classical.some (exists_reduced_fraction A x) | def | is_fraction_ring.num | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"num"
] | `f.num x` is the numerator of `x : f.codomain` as a reduced fraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
denom (x : K) : non_zero_divisors A | classical.some (classical.some_spec (exists_reduced_fraction A x)) | def | is_fraction_ring.denom | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"non_zero_divisors"
] | `f.num x` is the denominator of `x : f.codomain` as a reduced fraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
num_denom_reduced (x : K) {d} : d ∣ num A x → d ∣ denom A x → is_unit d | (classical.some_spec (classical.some_spec (exists_reduced_fraction A x))).1 | lemma | is_fraction_ring.num_denom_reduced | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"is_unit",
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_num_denom (x : K) : mk' K (num A x) (denom A x) = x | (classical.some_spec (classical.some_spec (exists_reduced_fraction A x))).2 | lemma | is_fraction_ring.mk'_num_denom | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"mk'",
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_mul_denom_eq_num_iff_eq {x y : K} :
x * algebra_map A K (denom A y) = algebra_map A K (num A y) ↔ x = y | ⟨λ h, by simpa only [mk'_num_denom] using eq_mk'_iff_mul_eq.mpr h,
λ h, eq_mk'_iff_mul_eq.mp (by rw [h, mk'_num_denom])⟩ | lemma | is_fraction_ring.num_mul_denom_eq_num_iff_eq | ring_theory.localization | src/ring_theory/localization/num_denom.lean | [
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.unique_factorization_domain"
] | [
"algebra_map",
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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