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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
formally_unramified.ext' [formally_unramified R A]
{C : Type u} [comm_ring C] (f : B →+* C) (hf : is_nilpotent f.ker)
(g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) : g₁ = g₂ | formally_unramified.lift_unique_of_ring_hom f hf g₁ g₂ (ring_hom.ext h) | lemma | algebra.formally_unramified.ext' | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"comm_ring",
"is_nilpotent",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.lift_unique' [formally_unramified R A]
{C : Type u} [comm_ring C] [algebra R C] (f : B →ₐ[R] C) (hf : is_nilpotent (f : B →+* C).ker)
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ | formally_unramified.ext' _ hf g₁ g₂ (alg_hom.congr_fun h) | lemma | algebra.formally_unramified.lift_unique' | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.congr_fun",
"algebra",
"comm_ring",
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.exists_lift {B : Type u} [comm_ring B] [_RB : algebra R B]
[formally_smooth R A] (I : ideal B)
(hI : is_nilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (ideal.quotient.mkₐ R I).comp f = g | begin
revert g,
change function.surjective (ideal.quotient.mkₐ R I).comp,
unfreezingI { revert _RB },
apply ideal.is_nilpotent.induction_on I hI,
{ introsI B _ I hI _, exact formally_smooth.comp_surjective I hI },
{ introsI B _ I J hIJ h₁ h₂ _ g,
let : ((B ⧸ I) ⧸ J.map (ideal.quotient.mk I)) ≃ₐ[R] B ⧸ J... | lemma | algebra.formally_smooth.exists_lift | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_equiv.comp_symm",
"alg_equiv.to_alg_hom_eq_coe",
"alg_hom.comp_assoc",
"alg_hom.id_comp",
"algebra",
"comm_ring",
"double_quot.quot_quot_equiv_quot_sup",
"ideal",
"ideal.is_nilpotent.induction_on",
"ideal.quot_equiv_of_eq",
"ideal.quotient.mk",
"ideal.quotient.mkₐ",
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.lift [formally_smooth R A] (I : ideal B)
(hI : is_nilpotent I) (g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B | (formally_smooth.exists_lift I hI g).some | def | algebra.formally_smooth.lift | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"ideal",
"is_nilpotent"
] | For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formally_smooth.comp_lift [formally_smooth R A] (I : ideal B)
(hI : is_nilpotent I) (g : A →ₐ[R] B ⧸ I) :
(ideal.quotient.mkₐ R I).comp (formally_smooth.lift I hI g) = g | (formally_smooth.exists_lift I hI g).some_spec | lemma | algebra.formally_smooth.comp_lift | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"ideal",
"ideal.quotient.mkₐ",
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.mk_lift [formally_smooth R A] (I : ideal B)
(hI : is_nilpotent I) (g : A →ₐ[R] B ⧸ I) (x : A) :
ideal.quotient.mk I (formally_smooth.lift I hI g x) = g x | alg_hom.congr_fun (formally_smooth.comp_lift I hI g : _) x | lemma | algebra.formally_smooth.mk_lift | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.congr_fun",
"ideal",
"ideal.quotient.mk",
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.lift_of_surjective [formally_smooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : function.surjective g) (hg' : is_nilpotent (g : B →+* C).ker) : A →ₐ[R] B | formally_smooth.lift _ hg'
((ideal.quotient_ker_alg_equiv_of_surjective hg).symm.to_alg_hom.comp f) | def | algebra.formally_smooth.lift_of_surjective | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"ideal.quotient_ker_alg_equiv_of_surjective",
"is_nilpotent"
] | For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formally_smooth.lift_of_surjective_apply [formally_smooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : function.surjective g) (hg' : is_nilpotent (g : B →+* C).ker) (x : A) :
g (formally_smooth.lift_of_surjective f g hg hg' x) = f x | begin
apply (ideal.quotient_ker_alg_equiv_of_surjective hg).symm.injective,
change _ = ((ideal.quotient_ker_alg_equiv_of_surjective hg).symm.to_alg_hom.comp f) x,
rw [← formally_smooth.mk_lift _ hg'
((ideal.quotient_ker_alg_equiv_of_surjective hg).symm.to_alg_hom.comp f)],
apply (ideal.quotient_ker_alg_equi... | lemma | algebra.formally_smooth.lift_of_surjective_apply | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_equiv.apply_symm_apply",
"ideal.ker_lift_alg_mk",
"ideal.quotient_ker_alg_equiv_of_right_inverse.apply",
"ideal.quotient_ker_alg_equiv_of_surjective",
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.comp_lift_of_surjective [formally_smooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : function.surjective g) (hg' : is_nilpotent (g : B →+* C).ker) :
g.comp (formally_smooth.lift_of_surjective f g hg hg') = f | alg_hom.ext (formally_smooth.lift_of_surjective_apply f g hg hg') | lemma | algebra.formally_smooth.comp_lift_of_surjective | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.ext",
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.of_equiv [formally_smooth R A] (e : A ≃ₐ[R] B) : formally_smooth R B | begin
constructor,
introsI C _ _ I hI f,
use (formally_smooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm,
rw [← alg_hom.comp_assoc, formally_smooth.comp_lift, alg_hom.comp_assoc, alg_equiv.comp_symm,
alg_hom.comp_id],
end | lemma | algebra.formally_smooth.of_equiv | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_equiv.comp_symm",
"alg_hom.comp_assoc",
"alg_hom.comp_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.of_equiv [formally_unramified R A] (e : A ≃ₐ[R] B) :
formally_unramified R B | begin
constructor,
introsI C _ _ I hI f₁ f₂ e',
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← alg_hom.comp_assoc, ← alg_hom.comp_assoc],
congr' 1,
refine formally_unramified.comp_injective I hI _,
rw [← alg_hom.comp_assoc, e', alg_hom.comp_assoc],
end | lemma | algebra.formally_unramified.of_equiv | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.comp_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_etale.of_equiv [formally_etale R A] (e : A ≃ₐ[R] B) : formally_etale R B | formally_etale.iff_unramified_and_smooth.mpr
⟨formally_unramified.of_equiv e, formally_smooth.of_equiv e⟩ | lemma | algebra.formally_etale.of_equiv | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.mv_polynomial (σ : Type u) : formally_smooth R (mv_polynomial σ R) | begin
constructor,
introsI C _ _ I hI f,
have : ∀ (s : σ), ∃ c : C, ideal.quotient.mk I c = f (mv_polynomial.X s),
{ exact λ s, ideal.quotient.mk_surjective _ },
choose g hg,
refine ⟨mv_polynomial.aeval g, _⟩,
ext s,
rw [← hg, alg_hom.comp_apply, mv_polynomial.aeval_X],
refl,
end | instance | algebra.formally_smooth.mv_polynomial | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.comp_apply",
"ideal.quotient.mk",
"ideal.quotient.mk_surjective",
"mv_polynomial",
"mv_polynomial.X",
"mv_polynomial.aeval_X"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.polynomial : formally_smooth R R[X] | @@formally_smooth.of_equiv _ _ _ _ _
(formally_smooth.mv_polynomial R punit) (mv_polynomial.punit_alg_equiv R) | instance | algebra.formally_smooth.polynomial | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"mv_polynomial.punit_alg_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.comp [formally_smooth R A] [formally_smooth A B] :
formally_smooth R B | begin
constructor,
introsI C _ _ I hI f,
obtain ⟨f', e⟩ := formally_smooth.comp_surjective I hI
(f.comp (is_scalar_tower.to_alg_hom R A B)),
letI := f'.to_ring_hom.to_algebra,
obtain ⟨f'', e'⟩ := formally_smooth.comp_surjective I hI
{ commutes' := alg_hom.congr_fun e.symm, ..f.to_ring_hom },
apply_f... | lemma | algebra.formally_smooth.comp | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.congr_fun",
"alg_hom.ext",
"alg_hom.restrict_scalars",
"is_scalar_tower.to_alg_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.comp [formally_unramified R A] [formally_unramified A B] :
formally_unramified R B | begin
constructor,
introsI C _ _ I hI f₁ f₂ e,
have e' := formally_unramified.lift_unique I ⟨2, hI⟩ (f₁.comp $ is_scalar_tower.to_alg_hom R A B)
(f₂.comp $ is_scalar_tower.to_alg_hom R A B)
(by rw [← alg_hom.comp_assoc, e, alg_hom.comp_assoc]),
letI := (f₁.comp (is_scalar_tower.to_alg_hom R A B)).to_rin... | lemma | algebra.formally_unramified.comp | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.comp_assoc",
"alg_hom.congr_fun",
"is_scalar_tower.to_alg_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.of_comp [formally_unramified R B] :
formally_unramified A B | begin
constructor,
introsI Q _ _ I e f₁ f₂ e',
letI := ((algebra_map A Q).comp (algebra_map R A)).to_algebra,
letI : is_scalar_tower R A Q := is_scalar_tower.of_algebra_map_eq' rfl,
refine alg_hom.restrict_scalars_injective R _,
refine formally_unramified.ext I ⟨2, e⟩ _,
intro x,
exact alg_hom.congr_fun... | lemma | algebra.formally_unramified.of_comp | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.congr_fun",
"alg_hom.restrict_scalars_injective",
"algebra_map",
"is_scalar_tower",
"is_scalar_tower.of_algebra_map_eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_etale.comp [formally_etale R A] [formally_etale A B] : formally_etale R B | formally_etale.iff_unramified_and_smooth.mpr
⟨formally_unramified.comp R A B, formally_smooth.comp R A B⟩ | lemma | algebra.formally_etale.comp | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.of_split [formally_smooth R P] (g : A →ₐ[R] P ⧸ f.to_ring_hom.ker ^ 2)
(hg : f.ker_square_lift.comp g = alg_hom.id R A) :
formally_smooth R A | begin
constructor,
introsI C _ _ I hI i,
let l : P ⧸ f.to_ring_hom.ker ^ 2 →ₐ[R] C,
{ refine ideal.quotient.liftₐ _ (formally_smooth.lift I ⟨2, hI⟩ (i.comp f)) _,
have : ring_hom.ker f ≤ I.comap (formally_smooth.lift I ⟨2, hI⟩ (i.comp f)),
{ rintros x (hx : f x = 0),
have : _ = i (f x) := (formall... | lemma | algebra.formally_smooth.of_split | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.coe_ring_hom_injective",
"alg_hom.comp_assoc",
"alg_hom.comp_id",
"alg_hom.id",
"ideal.le_comap_pow",
"ideal.pow_mono",
"ideal.quotient.liftₐ",
"ideal.quotient.mk_eq_mk",
"ideal.quotient.mkₐ",
"ideal.quotient.ring_hom_ext",
"ring_hom.ker",
"submodule.quotient.mk_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.iff_split_surjection [formally_smooth R P] :
formally_smooth R A ↔ ∃ g, f.ker_square_lift.comp g = alg_hom.id R A | begin
split,
{ introI,
have surj : function.surjective f.ker_square_lift :=
λ x, ⟨submodule.quotient.mk (hf x).some, (hf x).some_spec⟩,
have sqz : ring_hom.ker f.ker_square_lift.to_ring_hom ^ 2 = 0,
{ rw [alg_hom.ker_ker_sqare_lift, ideal.cotangent_ideal_square, ideal.zero_eq_bot] },
refine ⟨f... | lemma | algebra.formally_smooth.iff_split_surjection | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_equiv.apply_symm_apply",
"alg_hom.id",
"alg_hom.id_apply",
"alg_hom.ker_ker_sqare_lift",
"ideal.cotangent_ideal_square",
"ideal.quotient.mk_surjective",
"ideal.quotient_ker_alg_equiv_of_surjective",
"ideal.zero_eq_bot",
"ring_hom.ker"
] | Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formally_unramified.subsingleton_kaehler_differential [formally_unramified R S] :
subsingleton Ω[S⁄R] | begin
rw ← not_nontrivial_iff_subsingleton,
introsI h,
obtain ⟨f₁, f₂, e⟩ := (kaehler_differential.End_equiv R S).injective.nontrivial,
apply e,
ext1,
apply formally_unramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm),
rw [← alg_hom.to_ring_hom_eq_coe, alg_hom.ker_ker_sqare_lift],
exact ⟨_, ideal.cot... | instance | algebra.formally_unramified.subsingleton_kaehler_differential | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.ker_ker_sqare_lift",
"alg_hom.to_ring_hom_eq_coe",
"ideal.cotangent_ideal_square",
"kaehler_differential.End_equiv",
"not_nontrivial_iff_subsingleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.iff_subsingleton_kaehler_differential :
formally_unramified R S ↔ subsingleton Ω[S⁄R] | begin
split,
{ introsI, apply_instance },
{ introI H,
constructor,
introsI B _ _ I hI f₁ f₂ e,
letI := f₁.to_ring_hom.to_algebra,
haveI := is_scalar_tower.of_algebra_map_eq' (f₁.comp_algebra_map).symm,
have := ((kaehler_differential.linear_map_equiv_derivation R S).to_equiv.trans
(deriva... | lemma | algebra.formally_unramified.iff_subsingleton_kaehler_differential | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"derivation_to_square_zero_equiv_lift",
"is_scalar_tower.of_algebra_map_eq'",
"kaehler_differential.linear_map_equiv_derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.base_change [formally_unramified R A] :
formally_unramified B (B ⊗[R] A) | begin
constructor,
introsI C _ _ I hI f₁ f₂ e,
letI := ((algebra_map B C).comp (algebra_map R B)).to_algebra,
haveI : is_scalar_tower R B C := is_scalar_tower.of_algebra_map_eq' rfl,
apply alg_hom.restrict_scalars_injective R,
apply tensor_product.ext,
any_goals { apply_instance },
intros b a,
have : ... | instance | algebra.formally_unramified.base_change | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.congr_fun",
"alg_hom.restrict_scalars_apply",
"alg_hom.restrict_scalars_injective",
"algebra_map",
"is_scalar_tower",
"is_scalar_tower.of_algebra_map_eq'",
"mul_one",
"smul_eq_mul",
"tensor_product.ext",
"tensor_product.smul_tmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.base_change [formally_smooth R A] :
formally_smooth B (B ⊗[R] A) | begin
constructor,
introsI C _ _ I hI f,
letI := ((algebra_map B C).comp (algebra_map R B)).to_algebra,
haveI : is_scalar_tower R B C := is_scalar_tower.of_algebra_map_eq' rfl,
refine ⟨tensor_product.product_left_alg_hom (algebra.of_id B C) _, _⟩,
{ exact formally_smooth.lift I ⟨2, hI⟩
((f.restrict_sc... | instance | algebra.formally_smooth.base_change | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.restrict_scalars_injective",
"algebra.of_id",
"algebra.of_id_apply",
"algebra.smul_def",
"algebra_map",
"is_scalar_tower",
"is_scalar_tower.of_algebra_map_eq'",
"mul_one",
"smul_eq_mul",
"tensor_product.ext",
"tensor_product.smul_tmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_etale.base_change [formally_etale R A] :
formally_etale B (B ⊗[R] A) | formally_etale.iff_unramified_and_smooth.mpr ⟨infer_instance, infer_instance⟩ | instance | algebra.formally_etale.base_change | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.of_is_localization : formally_smooth R Rₘ | begin
constructor,
introsI Q _ _ I e f,
have : ∀ x : M, is_unit (algebra_map R Q x),
{ intro x,
apply (is_nilpotent.is_unit_quotient_mk_iff ⟨2, e⟩).mp,
convert (is_localization.map_units Rₘ x).map f,
simp only [ideal.quotient.mk_algebra_map, alg_hom.commutes] },
let : Rₘ →ₐ[R] Q := { commutes' := ... | lemma | algebra.formally_smooth.of_is_localization | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.coe_ring_hom_injective",
"alg_hom.commutes",
"algebra_map",
"ideal.quotient.mk_algebra_map",
"is_localization.lift",
"is_localization.lift_eq",
"is_localization.ring_hom_ext",
"is_nilpotent.is_unit_quotient_mk_iff",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.of_is_localization : formally_unramified R Rₘ | begin
constructor,
introsI Q _ _ I e f₁ f₂ e,
apply alg_hom.coe_ring_hom_injective,
refine is_localization.ring_hom_ext M _,
ext,
simp,
end | lemma | algebra.formally_unramified.of_is_localization | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.coe_ring_hom_injective",
"is_localization.ring_hom_ext"
] | This holds in general for epimorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formally_etale.of_is_localization : formally_etale R Rₘ | formally_etale.iff_unramified_and_smooth.mpr
⟨formally_unramified.of_is_localization M, formally_smooth.of_is_localization M⟩ | lemma | algebra.formally_etale.of_is_localization | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.localization_base [formally_smooth R Sₘ] : formally_smooth Rₘ Sₘ | begin
constructor,
introsI Q _ _ I e f,
letI := ((algebra_map Rₘ Q).comp (algebra_map R Rₘ)).to_algebra,
letI : is_scalar_tower R Rₘ Q := is_scalar_tower.of_algebra_map_eq' rfl,
let f : Sₘ →ₐ[Rₘ] Q,
{ refine { commutes' := _, ..(formally_smooth.lift I ⟨2, e⟩ (f.restrict_scalars R)) },
intro r,
chang... | lemma | algebra.formally_smooth.localization_base | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.comp_algebra_map",
"algebra_map",
"is_localization.ring_hom_ext",
"is_scalar_tower",
"is_scalar_tower.algebra_map_eq",
"is_scalar_tower.of_algebra_map_eq'",
"ring_hom.comp",
"ring_hom.comp_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.localization_base [formally_unramified R Sₘ] :
formally_unramified Rₘ Sₘ | formally_unramified.of_comp R Rₘ Sₘ | lemma | algebra.formally_unramified.localization_base | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [] | This actually does not need the localization instance, and is stated here again for
consistency. See `algebra.formally_unramified.of_comp` instead.
The intended use is for copying proofs between `formally_{unramified, smooth, etale}`
without the need to change anything (including removing redundant arguments). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formally_etale.localization_base [formally_etale R Sₘ] : formally_etale Rₘ Sₘ | formally_etale.iff_unramified_and_smooth.mpr
⟨formally_unramified.localization_base M, formally_smooth.localization_base M⟩ | lemma | algebra.formally_etale.localization_base | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_smooth.localization_map [formally_smooth R S] : formally_smooth Rₘ Sₘ | begin
haveI : formally_smooth S Sₘ := formally_smooth.of_is_localization (M.map (algebra_map R S)),
haveI : formally_smooth R Sₘ := formally_smooth.comp R S Sₘ,
exact formally_smooth.localization_base M
end | lemma | algebra.formally_smooth.localization_map | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.localization_map [formally_unramified R S] : formally_unramified Rₘ Sₘ | begin
haveI : formally_unramified S Sₘ :=
formally_unramified.of_is_localization (M.map (algebra_map R S)),
haveI : formally_unramified R Sₘ := formally_unramified.comp R S Sₘ,
exact formally_unramified.localization_base M
end | lemma | algebra.formally_unramified.localization_map | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_etale.localization_map [formally_etale R S] : formally_etale Rₘ Sₘ | begin
haveI : formally_etale S Sₘ := formally_etale.of_is_localization (M.map (algebra_map R S)),
haveI : formally_etale R Sₘ := formally_etale.comp R S Sₘ,
exact formally_etale.localization_base M
end | lemma | algebra.formally_etale.localization_map | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_ne_zero_of_left (hp : p ≠ 0) : gcd_monoid.gcd p q ≠ 0 | λ h, hp $ eq_zero_of_zero_dvd (h ▸ gcd_dvd_left p q) | lemma | gcd_ne_zero_of_left | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"eq_zero_of_zero_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_ne_zero_of_right (hp : q ≠ 0) : gcd_monoid.gcd p q ≠ 0 | λ h, hp $ eq_zero_of_zero_dvd (h ▸ gcd_dvd_right p q) | lemma | gcd_ne_zero_of_right | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"eq_zero_of_zero_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) :
p / gcd_monoid.gcd p q ≠ 0 | begin
obtain ⟨r, hr⟩ := gcd_monoid.gcd_dvd_left p q,
obtain ⟨pq0, r0⟩ : gcd_monoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp),
rw [hr, mul_comm, mul_div_cancel _ pq0] { occs := occurrences.pos [1] },
exact r0,
end | lemma | left_div_gcd_ne_zero | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"mul_comm",
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) :
q / gcd_monoid.gcd p q ≠ 0 | begin
obtain ⟨r, hr⟩ := gcd_monoid.gcd_dvd_right p q,
obtain ⟨pq0, r0⟩ : gcd_monoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq),
rw [hr, mul_comm, mul_div_cancel _ pq0] { occs := occurrences.pos [1] },
exact r0,
end | lemma | right_div_gcd_ne_zero | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"mul_comm",
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_div_gcd_div_gcd (hq : q ≠ 0) :
is_coprime (p / gcd_monoid.gcd p q) (q / gcd_monoid.gcd p q) | (gcd_is_unit_iff _ _).1 $ is_unit_gcd_of_eq_mul_gcd
(euclidean_domain.mul_div_cancel' (gcd_ne_zero_of_right hq) $ gcd_dvd_left _ _).symm
(euclidean_domain.mul_div_cancel' (gcd_ne_zero_of_right hq) $ gcd_dvd_right _ _).symm $
gcd_ne_zero_of_right hq | lemma | is_coprime_div_gcd_div_gcd | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"euclidean_domain.mul_div_cancel'",
"gcd_is_unit_iff",
"gcd_ne_zero_of_right",
"is_coprime",
"is_unit_gcd_of_eq_mul_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_monoid (R) [euclidean_domain R] : gcd_monoid R | { gcd := gcd,
lcm := lcm,
gcd_dvd_left := gcd_dvd_left,
gcd_dvd_right := gcd_dvd_right,
dvd_gcd := λ a b c, dvd_gcd,
gcd_mul_lcm := λ a b, by rw euclidean_domain.gcd_mul_lcm,
lcm_zero_left := lcm_zero_left,
lcm_zero_right := lcm_zero_right } | def | euclidean_domain.gcd_monoid | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"euclidean_domain",
"euclidean_domain.gcd_mul_lcm",
"gcd_monoid",
"gcd_mul_lcm"
] | Create a `gcd_monoid` whose `gcd_monoid.gcd` matches `euclidean_domain.gcd`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_gcd {α} [euclidean_domain α] (x y : α) :
span ({gcd x y} : set α) = span ({x, y} : set α) | begin
letI := euclidean_domain.gcd_monoid α,
exact span_gcd x y,
end | theorem | euclidean_domain.span_gcd | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"euclidean_domain",
"euclidean_domain.gcd_monoid",
"span_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_is_unit_iff {α} [euclidean_domain α] {x y : α} :
is_unit (gcd x y) ↔ is_coprime x y | begin
letI := euclidean_domain.gcd_monoid α,
exact gcd_is_unit_iff x y,
end | theorem | euclidean_domain.gcd_is_unit_iff | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"euclidean_domain",
"euclidean_domain.gcd_monoid",
"gcd_is_unit_iff",
"is_coprime",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_of_dvd {α} [euclidean_domain α] {x y : α}
(nonzero : ¬ (x = 0 ∧ y = 0)) (H : ∀ z ∈ nonunits α, z ≠ 0 → z ∣ x → ¬ z ∣ y) :
is_coprime x y | begin
letI := euclidean_domain.gcd_monoid α,
exact is_coprime_of_dvd x y nonzero H,
end | theorem | euclidean_domain.is_coprime_of_dvd | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"euclidean_domain",
"euclidean_domain.gcd_monoid",
"is_coprime",
"is_coprime_of_dvd",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_or_coprime {α} [euclidean_domain α] (x y : α)
(h : irreducible x) : x ∣ y ∨ is_coprime x y | begin
letI := euclidean_domain.gcd_monoid α,
exact dvd_or_coprime x y h,
end | theorem | euclidean_domain.dvd_or_coprime | ring_theory | src/ring_theory/euclidean_domain.lean | [
"algebra.gcd_monoid.basic",
"algebra.euclidean_domain.basic",
"ring_theory.ideal.basic",
"ring_theory.principal_ideal_domain"
] | [
"dvd_or_coprime",
"euclidean_domain",
"euclidean_domain.gcd_monoid",
"irreducible",
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.filtration (M : Type u) [add_comm_group M] [module R M] | (N : ℕ → submodule R M)
(mono : ∀ i, N (i + 1) ≤ N i)
(smul_le : ∀ i, I • N i ≤ N (i + 1)) | structure | ideal.filtration | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"add_comm_group",
"module",
"submodule"
] | An `I`-filtration on the module `M` is a sequence of decreasing submodules `N i` such that
`I • (N i) ≤ N (i + 1)`. Note that we do not require the filtration to start from `⊤`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) | begin
induction i,
{ simp },
{ rw [pow_succ, mul_smul, nat.succ_eq_add_one, add_assoc, add_comm 1, ← add_assoc],
exact (submodule.smul_mono_right i_ih).trans (F.smul_le _) }
end | lemma | ideal.filtration.pow_smul_le | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"pow_succ",
"submodule.smul_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_smul_le_pow_smul (i j k : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • F.N (i + j) | by { rw [add_comm, pow_add, mul_smul], exact submodule.smul_mono_right (F.pow_smul_le i j) } | lemma | ideal.filtration.pow_smul_le_pow_smul | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"pow_add",
"submodule.smul_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone : antitone F.N | antitone_nat_of_succ_le F.mono | lemma | ideal.filtration.antitone | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"antitone",
"antitone_nat_of_succ_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ideal.trivial_filtration (I : ideal R) (N : submodule R M) : I.filtration M | { N := λ i, N,
mono := λ i, le_of_eq rfl,
smul_le := λ i, submodule.smul_le_right } | def | ideal.trivial_filtration | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"ideal",
"submodule",
"submodule.smul_le_right"
] | The trivial `I`-filtration of `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_N : (F ⊔ F').N = F.N ⊔ F'.N | rfl | lemma | ideal.filtration.sup_N | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_N (S : set (I.filtration M)) : (Sup S).N = Sup (ideal.filtration.N '' S) | rfl | lemma | ideal.filtration.Sup_N | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_N : (F ⊓ F').N = F.N ⊓ F'.N | rfl | lemma | ideal.filtration.inf_N | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_N (S : set (I.filtration M)) : (Inf S).N = Inf (ideal.filtration.N '' S) | rfl | lemma | ideal.filtration.Inf_N | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_N : (⊤ : I.filtration M).N = ⊤ | rfl | lemma | ideal.filtration.top_N | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_N : (⊥ : I.filtration M).N = ⊥ | rfl | lemma | ideal.filtration.bot_N | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_N {ι : Sort*} (f : ι → I.filtration M) : (supr f).N = ⨆ i, (f i).N | congr_arg Sup (set.range_comp _ _).symm | lemma | ideal.filtration.supr_N | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"set.range_comp",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_N {ι : Sort*} (f : ι → I.filtration M) : (infi f).N = ⨅ i, (f i).N | congr_arg Inf (set.range_comp _ _).symm | lemma | ideal.filtration.infi_N | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"infi",
"set.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable : Prop | ∃ n₀, ∀ n ≥ n₀, I • F.N n = F.N (n + 1) | def | ideal.filtration.stable | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [] | An `I` filtration is stable if `I • F.N n = F.N (n+1)` for large enough `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.ideal.stable_filtration (I : ideal R) (N : submodule R M) :
I.filtration M | { N := λ i, I ^ i • N,
mono := λ i, by { rw [add_comm, pow_add, mul_smul], exact submodule.smul_le_right },
smul_le := λ i, by { rw [add_comm, pow_add, mul_smul, pow_one], exact le_refl _ } } | def | ideal.stable_filtration | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"ideal",
"pow_add",
"pow_one",
"submodule",
"submodule.smul_le_right"
] | The trivial stable `I`-filtration of `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.ideal.stable_filtration_stable (I : ideal R) (N : submodule R M) :
(I.stable_filtration N).stable | by { use 0, intros n _, dsimp, rw [add_comm, pow_add, mul_smul, pow_one] } | lemma | ideal.stable_filtration_stable | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"ideal",
"pow_add",
"pow_one",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable.exists_pow_smul_eq :
∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ | begin
obtain ⟨n₀, hn⟩ := h,
use n₀,
intro k,
induction k,
{ simp },
{ rw [nat.succ_eq_add_one, ← add_assoc, ← hn, k_ih, add_comm, pow_add, mul_smul, pow_one],
linarith }
end | lemma | ideal.filtration.stable.exists_pow_smul_eq | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"pow_add",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable.exists_pow_smul_eq_of_ge :
∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | begin
obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq,
use n₀,
intros n hn,
convert hn₀ (n - n₀),
rw [add_comm, tsub_add_cancel_of_le hn],
end | lemma | ideal.filtration.stable.exists_pow_smul_eq_of_ge | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_iff_exists_pow_smul_eq_of_ge :
F.stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ | begin
refine ⟨stable.exists_pow_smul_eq_of_ge, λ h, ⟨h.some, λ n hn, _⟩⟩,
rw [h.some_spec n hn, h.some_spec (n+1) (by linarith), smul_smul, ← pow_succ,
tsub_add_eq_add_tsub hn],
end | lemma | ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"pow_succ",
"smul_smul",
"tsub_add_eq_add_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable.exists_forall_le (h : F.stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n | begin
obtain ⟨n₀, hF⟩ := h,
use n₀,
intro n,
induction n with n hn,
{ refine (F.antitone _).trans e, simp },
{ rw [nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, ← hF],
exact (submodule.smul_mono_right hn).trans (F'.smul_le _),
simp },
end | lemma | ideal.filtration.stable.exists_forall_le | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"nat.succ_eq_one_add",
"submodule.smul_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable.bounded_difference (h : F.stable) (h' : F'.stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n | begin
obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e),
obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm),
use max n₁ n₂,
intro n,
refine ⟨(F.antitone _).trans (h₁ n), (F'.antitone _).trans (h₂ n)⟩; simp
end | lemma | ideal.filtration.stable.bounded_difference | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule : submodule (rees_algebra I) (polynomial_module R M) | { carrier := { f | ∀ i, f i ∈ F.N i },
add_mem' := λ f g hf hg i, submodule.add_mem _ (hf i) (hg i),
zero_mem' := λ i, submodule.zero_mem _,
smul_mem' := λ r f hf i, begin
rw [subalgebra.smul_def, polynomial_module.smul_apply],
apply submodule.sum_mem,
rintro ⟨j, k⟩ e,
rw finset.nat.mem_antidiagon... | def | ideal.filtration.submodule | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"finset.nat.mem_antidiagonal",
"polynomial_module",
"polynomial_module.smul_apply",
"rees_algebra",
"subalgebra.smul_def",
"submodule",
"submodule.add_mem",
"submodule.smul_mem_smul",
"submodule.sum_mem",
"submodule.zero_mem"
] | The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_submodule (f : polynomial_module R M) : f ∈ F.submodule ↔ ∀ i, f i ∈ F.N i | iff.rfl | lemma | ideal.filtration.mem_submodule | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"polynomial_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ F'.submodule | by { ext, exact forall_and_distrib } | lemma | ideal.filtration.inf_submodule | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"forall_and_distrib",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule_inf_hom :
inf_hom (I.filtration M) (submodule (rees_algebra I) (polynomial_module R M)) | { to_fun := ideal.filtration.submodule, map_inf' := inf_submodule } | def | ideal.filtration.submodule_inf_hom | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"ideal.filtration.submodule",
"inf_hom",
"polynomial_module",
"rees_algebra",
"submodule"
] | `ideal.filtration.submodule` as an `inf_hom` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule_closure_single :
add_submonoid.closure (⋃ i, single R i '' (F.N i : set M)) = F.submodule.to_add_submonoid | begin
apply le_antisymm,
{ rw [add_submonoid.closure_le, set.Union_subset_iff],
rintro i _ ⟨m, hm, rfl⟩ j,
rw single_apply,
split_ifs,
{ rwa ← h },
{ exact (F.N j).zero_mem } },
{ intros f hf,
rw [← f.sum_single],
apply add_submonoid.sum_mem _ _,
rintros c -,
exact add_submonoi... | lemma | ideal.filtration.submodule_closure_single | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"set.Union_subset_iff",
"set.mem_image_of_mem",
"set.subset_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule_span_single :
submodule.span (rees_algebra I) (⋃ i, single R i '' (F.N i : set M)) = F.submodule | begin
rw [← submodule.span_closure, submodule_closure_single],
simp,
end | lemma | ideal.filtration.submodule_span_single | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"rees_algebra",
"submodule.span",
"submodule.span_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | begin
rw [← submodule_span_single, ← has_le.le.le_iff_eq, submodule.span_le,
set.Union_subset_iff],
swap, { exact submodule.span_mono (set.Union₂_subset_Union _ _) },
split,
{ intros H n hn,
refine (F.smul_le n).antisymm _,
intros x hx,
obtain ⟨l, hl⟩ := (finsupp.mem_span_iff_total _ _ _).mp (H ... | lemma | ideal.filtration.submodule_eq_span_le_iff_stable_ge | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"add_tsub_assoc_of_le",
"finsupp.mem_span_iff_total",
"finsupp.single_eq_same",
"finsupp.sum_apply",
"finsupp.total_apply",
"has_le.le.le_iff_eq",
"nat.lt_succ_iff",
"pow_one",
"rees_algebra",
"set.Union_subset_iff",
"set.Union₂_subset_Union",
"set.mem_image_of_mem",
"set.subset.trans",
"s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule_fg_iff_stable (hF' : ∀ i, (F.N i).fg) :
F.submodule.fg ↔ F.stable | begin
classical,
delta ideal.filtration.stable,
simp_rw ← F.submodule_eq_span_le_iff_stable_ge,
split,
{ rintro H,
apply H.stablizes_of_supr_eq
⟨λ n₀, submodule.span _ (⋃ (i : ℕ) (H : i ≤ n₀), single R i '' ↑(F.N i)), _⟩,
{ dsimp,
rw [← submodule.span_Union, ← submodule_span_single],
... | lemma | ideal.filtration.submodule_fg_iff_stable | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"exists_prop",
"finset.coe_image",
"finset.mem_range_succ_iff",
"ideal.filtration.stable",
"polynomial_module",
"rees_algebra",
"set.Union₂_subset_iff",
"set.mem_Union",
"set.mem_image",
"set.subset.trans",
"set.subset_Union₂",
"set_like.mem_coe",
"submodule.fg_supr",
"submodule.map_span",... | If the components of a filtration are finitely generated, then the filtration is stable iff
its associated submodule of is finitely generated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stable.of_le [is_noetherian_ring R] [h : module.finite R M] (hF : F.stable)
{F' : I.filtration M} (hf : F' ≤ F) : F'.stable | begin
haveI := is_noetherian_of_fg_of_noetherian' h.1,
rw ← submodule_fg_iff_stable at hF ⊢,
any_goals { intro i, exact is_noetherian.noetherian _ },
have := is_noetherian_of_fg_of_noetherian _ hF,
rw is_noetherian_submodule at this,
exact this _ (order_hom_class.mono (submodule_inf_hom M I) hf),
end | lemma | ideal.filtration.stable.of_le | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"is_noetherian_of_fg_of_noetherian",
"is_noetherian_of_fg_of_noetherian'",
"is_noetherian_ring",
"is_noetherian_submodule",
"module.finite",
"order_hom_class.mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable.inter_right [is_noetherian_ring R] [h : module.finite R M] (hF : F.stable) :
(F ⊓ F').stable | hF.of_le inf_le_left | lemma | ideal.filtration.stable.inter_right | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"inf_le_left",
"is_noetherian_ring",
"module.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable.inter_left [is_noetherian_ring R] [h : module.finite R M] (hF : F.stable) :
(F' ⊓ F).stable | hF.of_le inf_le_right | lemma | ideal.filtration.stable.inter_left | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"inf_le_right",
"is_noetherian_ring",
"module.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.exists_pow_inf_eq_pow_smul [is_noetherian_ring R] [h : module.finite R M]
(N : submodule R M) : ∃ k : ℕ, ∀ n ≥ k, I ^ n • ⊤ ⊓ N = I ^ (n - k) • (I ^ k • ⊤ ⊓ N) | ((I.stable_filtration_stable ⊤).inter_right (I.trivial_filtration N)).exists_pow_smul_eq_of_ge | lemma | ideal.exists_pow_inf_eq_pow_smul | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"is_noetherian_ring",
"module.finite",
"submodule"
] | **Artin-Rees lemma** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.mem_infi_smul_pow_eq_bot_iff [is_noetherian_ring R] [hM : module.finite R M] (x : M) :
x ∈ (⨅ i : ℕ, I ^ i • ⊤ : submodule R M) ↔ ∃ r : I, (r : R) • x = x | begin
let N := (⨅ i : ℕ, I ^ i • ⊤ : submodule R M),
have hN : ∀ k, (I.stable_filtration ⊤ ⊓ I.trivial_filtration N).N k = N,
{ intro k, exact inf_eq_right.mpr ((infi_le _ k).trans $ le_of_eq $ by simp) },
split,
{ haveI := is_noetherian_of_fg_of_noetherian' hM.out,
obtain ⟨r, hr₁, hr₂⟩ := submodule.exist... | lemma | ideal.mem_infi_smul_pow_eq_bot_iff | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"infi_le",
"is_noetherian_of_fg_of_noetherian'",
"is_noetherian_ring",
"module.finite",
"nat.succ_eq_one_add",
"pow_add",
"pow_one",
"smul_smul",
"submodule",
"submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul",
"submodule.mem_infi",
"submodule.smul_mem_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.infi_pow_smul_eq_bot_of_local_ring [is_noetherian_ring R] [local_ring R]
[module.finite R M] (h : I ≠ ⊤) :
(⨅ i : ℕ, I ^ i • ⊤ : submodule R M) = ⊥ | begin
rw eq_bot_iff,
intros x hx,
obtain ⟨r, hr⟩ := (I.mem_infi_smul_pow_eq_bot_iff x).mp hx,
have := local_ring.is_unit_one_sub_self_of_mem_nonunits _ (local_ring.le_maximal_ideal h r.prop),
apply this.smul_left_cancel.mp,
swap, { apply_instance },
simp [sub_smul, hr],
end | lemma | ideal.infi_pow_smul_eq_bot_of_local_ring | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"eq_bot_iff",
"is_noetherian_ring",
"local_ring",
"local_ring.is_unit_one_sub_self_of_mem_nonunits",
"local_ring.le_maximal_ideal",
"module.finite",
"sub_smul",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.infi_pow_eq_bot_of_local_ring [is_noetherian_ring R] [local_ring R] (h : I ≠ ⊤) :
(⨅ i : ℕ, I ^ i) = ⊥ | begin
convert I.infi_pow_smul_eq_bot_of_local_ring h,
ext i,
rw [smul_eq_mul, ← ideal.one_eq_top, mul_one],
apply_instance,
end | lemma | ideal.infi_pow_eq_bot_of_local_ring | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"ideal.one_eq_top",
"is_noetherian_ring",
"local_ring",
"mul_one",
"smul_eq_mul"
] | **Krull's intersection theorem** for noetherian local rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.infi_pow_eq_bot_of_is_domain [is_noetherian_ring R] [is_domain R] (h : I ≠ ⊤) :
(⨅ i : ℕ, I ^ i) = ⊥ | begin
rw eq_bot_iff,
intros x hx,
by_contra hx',
have := ideal.mem_infi_smul_pow_eq_bot_iff I x,
simp_rw [smul_eq_mul, ← ideal.one_eq_top, mul_one] at this,
obtain ⟨r, hr⟩ := this.mp hx,
have := mul_right_cancel₀ hx' (hr.trans (one_mul x).symm),
exact I.eq_top_iff_one.not.mp h (this ▸ r.prop),
end | lemma | ideal.infi_pow_eq_bot_of_is_domain | ring_theory | src/ring_theory/filtration.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.noetherian",
"ring_theory.rees_algebra",
"ring_theory.finiteness",
"data.polynomial.module",
"order.hom.lattice"
] | [
"by_contra",
"eq_bot_iff",
"ideal.mem_infi_smul_pow_eq_bot_iff",
"ideal.one_eq_top",
"is_domain",
"is_noetherian_ring",
"mul_one",
"mul_right_cancel₀",
"one_mul",
"smul_eq_mul"
] | **Krull's intersection theorem** for noetherian domains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fg (N : submodule R M) : Prop | ∃ S : finset M, submodule.span R ↑S = N | def | submodule.fg | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finset",
"submodule",
"submodule.span"
] | A submodule of `M` is finitely generated if it is the span of a finite subset of `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fg_def {N : submodule R M} :
N.fg ↔ ∃ S : set M, S.finite ∧ span R S = N | ⟨λ ⟨t, h⟩, ⟨_, finset.finite_to_set t, h⟩, begin
rintro ⟨t', h, rfl⟩,
rcases finite.exists_finset_coe h with ⟨t, rfl⟩,
exact ⟨t, rfl⟩
end⟩ | theorem | submodule.fg_def | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finset.finite_to_set",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_iff_add_submonoid_fg (P : submodule ℕ M) :
P.fg ↔ P.to_add_submonoid.fg | ⟨λ ⟨S, hS⟩, ⟨S, by simpa [← span_nat_eq_add_submonoid_closure] using hS⟩,
λ ⟨S, hS⟩, ⟨S, by simpa [← span_nat_eq_add_submonoid_closure] using hS⟩⟩ | lemma | submodule.fg_iff_add_submonoid_fg | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_iff_add_subgroup_fg {G : Type*} [add_comm_group G] (P : submodule ℤ G) :
P.fg ↔ P.to_add_subgroup.fg | ⟨λ ⟨S, hS⟩, ⟨S, by simpa [← span_int_eq_add_subgroup_closure] using hS⟩,
λ ⟨S, hS⟩, ⟨S, by simpa [← span_int_eq_add_subgroup_closure] using hS⟩⟩ | lemma | submodule.fg_iff_add_subgroup_fg | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_group",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_iff_exists_fin_generating_family {N : submodule R M} :
N.fg ↔ ∃ (n : ℕ) (s : fin n → M), span R (range s) = N | begin
rw fg_def,
split,
{ rintros ⟨S, Sfin, hS⟩,
obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding,
exact ⟨n, f, hS⟩, },
{ rintros ⟨n, s, hs⟩,
refine ⟨range s, finite_range s, hs⟩ },
end | lemma | submodule.fg_iff_exists_fin_generating_family | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [comm_ring R]
{M : Type*} [add_comm_group M] [module R M]
(I : ideal R) (N : submodule R M) (hn : N.fg) (hin : N ≤ I • N) :
∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) | begin
rw fg_def at hn, rcases hn with ⟨s, hfs, hs⟩,
have : ∃ r : R, r - 1 ∈ I ∧ N ≤ (I • span R s).comap (linear_map.lsmul R M r) ∧ s ⊆ N,
{ refine ⟨1, _, _, _⟩,
{ rw sub_self, exact I.zero_mem },
{ rw [hs], intros n hn, rw [mem_comap], change (1:R) • n ∈ I • N, rw one_smul, exact hin hn },
{ rw [← sp... | theorem | submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_group",
"comm_ring",
"ideal",
"ih",
"linear_map.lsmul",
"module",
"mul_comm",
"mul_one",
"one_smul",
"set.finite.dinduction_on",
"set.insert_subset",
"set.singleton_union",
"smul_add",
"smul_smul",
"sub_smul",
"submodule"
] | **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_mem_and_smul_eq_self_of_fg_of_le_smul {R : Type*} [comm_ring R]
{M : Type*} [add_comm_group M] [module R M]
(I : ideal R) (N : submodule R M) (hn : N.fg) (hin : N ≤ I • N) :
∃ r ∈ I, ∀ n ∈ N, r • n = n | begin
obtain ⟨r, hr, hr'⟩ := N.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I hn hin,
exact ⟨-(r-1), I.neg_mem hr, λ n hn, by simpa [sub_smul] using hr' n hn⟩,
end | theorem | submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_group",
"comm_ring",
"ideal",
"module",
"sub_smul",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_bot : (⊥ : submodule R M).fg | ⟨∅, by rw [finset.coe_empty, span_empty]⟩ | theorem | submodule.fg_bot | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finset.coe_empty",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.subalgebra.fg_bot_to_submodule {R A : Type*}
[comm_semiring R] [semiring A] [algebra R A] :
(⊥ : subalgebra R A).to_submodule.fg | ⟨{1}, by simp [algebra.to_submodule_bot] ⟩ | lemma | subalgebra.fg_bot_to_submodule | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"algebra",
"algebra.to_submodule_bot",
"comm_semiring",
"semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_unit {R A : Type*} [comm_semiring R] [semiring A] [algebra R A]
(I : (submodule R A)ˣ) : (I : submodule R A).fg | begin
have : (1 : A) ∈ (I * ↑I⁻¹ : submodule R A),
{ rw I.mul_inv, exact one_le.mp le_rfl },
obtain ⟨T, T', hT, hT', one_mem⟩ := mem_span_mul_finite_of_mem_mul this,
refine ⟨T, span_eq_of_le _ hT _⟩,
rw [← one_mul ↑I, ← mul_one (span R ↑T)],
conv_rhs { rw [← I.inv_mul, ← mul_assoc] },
refine mul_le_mul_le... | lemma | submodule.fg_unit | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"algebra",
"comm_semiring",
"le_rfl",
"mul_assoc",
"mul_le_mul_left",
"mul_le_mul_right",
"mul_one",
"one_le",
"one_mul",
"semiring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_of_is_unit {R A : Type*} [comm_semiring R] [semiring A] [algebra R A]
{I : submodule R A} (hI : is_unit I) : I.fg | fg_unit hI.unit | lemma | submodule.fg_of_is_unit | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"algebra",
"comm_semiring",
"is_unit",
"semiring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_span {s : set M} (hs : s.finite) : fg (span R s) | ⟨hs.to_finset, by rw [hs.coe_to_finset]⟩ | theorem | submodule.fg_span | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_span_singleton (x : M) : fg (R ∙ x) | fg_span (finite_singleton x) | theorem | submodule.fg_span_singleton | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg.sup {N₁ N₂ : submodule R M}
(hN₁ : N₁.fg) (hN₂ : N₂.fg) : (N₁ ⊔ N₂).fg | let ⟨t₁, ht₁⟩ := fg_def.1 hN₁, ⟨t₂, ht₂⟩ := fg_def.1 hN₂ in
fg_def.2 ⟨t₁ ∪ t₂, ht₁.1.union ht₂.1, by rw [span_union, ht₁.2, ht₂.2]⟩ | theorem | submodule.fg.sup | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_finset_sup {ι : Type*} (s : finset ι) (N : ι → submodule R M) (h : ∀ i ∈ s, (N i).fg) :
(s.sup N).fg | finset.sup_induction fg_bot (λ a ha b hb, ha.sup hb) h | lemma | submodule.fg_finset_sup | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finset",
"finset.sup_induction",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_bsupr {ι : Type*} (s : finset ι) (N : ι → submodule R M) (h : ∀ i ∈ s, (N i).fg) :
(⨆ i ∈ s, N i).fg | by simpa only [finset.sup_eq_supr] using fg_finset_sup s N h | lemma | submodule.fg_bsupr | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finset",
"finset.sup_eq_supr",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_supr {ι : Type*} [finite ι] (N : ι → submodule R M) (h : ∀ i, (N i).fg) :
(supr N).fg | by { casesI nonempty_fintype ι, simpa using fg_bsupr finset.univ N (λ i hi, h i) } | lemma | submodule.fg_supr | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"finset.univ",
"nonempty_fintype",
"submodule",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg.map {N : submodule R M} (hs : N.fg) : (N.map f).fg | let ⟨t, ht⟩ := fg_def.1 hs in fg_def.2 ⟨f '' t, ht.1.image _, by rw [span_image, ht.2]⟩ | theorem | submodule.fg.map | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_of_fg_map_injective (f : M →ₗ[R] P) (hf : function.injective f) {N : submodule R M}
(hfn : (N.map f).fg) : N.fg | let ⟨t, ht⟩ := hfn in ⟨t.preimage f $ λ x _ y _ h, hf h,
submodule.map_injective_of_injective hf $ by { rw [f.map_span, finset.coe_preimage,
set.image_preimage_eq_inter_range, set.inter_eq_self_of_subset_left, ht],
rw [← linear_map.range_coe, ← span_le, ht, ← map_top], exact map_mono le_top }⟩ | lemma | submodule.fg_of_fg_map_injective | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finset.coe_preimage",
"le_top",
"linear_map.range_coe",
"set.image_preimage_eq_inter_range",
"set.inter_eq_self_of_subset_left",
"submodule",
"submodule.map_injective_of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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