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_jacobson {R : Type*} [comm_ring R] {ι : Type*} [finite ι] [is_jacobson R] :
is_jacobson (mv_polynomial ι R) | begin
casesI nonempty_fintype ι,
haveI := classical.dec_eq ι,
let e := fintype.equiv_fin ι,
rw is_jacobson_iso (rename_equiv R e).to_ring_equiv,
exact is_jacobson_mv_polynomial_fin _
end | instance | ideal.mv_polynomial.is_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"classical.dec_eq",
"comm_ring",
"finite",
"fintype.equiv_fin",
"mv_polynomial",
"nonempty_fintype"
] | General form of the nullstellensatz for Jacobson rings, since in a Jacobson ring we have
`Inf {P maximal | P ≥ I} = Inf {P prime | P ≥ I} = I.radical`. Fields are always Jacobson,
and in that special case this is (most of) the classical Nullstellensatz,
since `I(V(I))` is the intersection of maximal ideals contai... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_mk_comp_C_is_integral_of_jacobson
{R : Type*} [comm_ring R] [is_jacobson R]
(P : ideal (mv_polynomial (fin n) R)) [P.is_maximal] :
((quotient.mk P).comp mv_polynomial.C : R →+* mv_polynomial _ R ⧸ P).is_integral | begin
unfreezingI {induction n with n IH},
{ refine ring_hom.is_integral_of_surjective _ (function.surjective.comp quotient.mk_surjective _),
exact C_surjective (fin 0) },
{ rw [← fin_succ_equiv_comp_C_eq_C, ← ring_hom.comp_assoc, ← ring_hom.comp_assoc,
← quotient_map_comp_mk le_rfl, ring_hom.comp_assoc... | lemma | ideal.mv_polynomial.quotient_mk_comp_C_is_integral_of_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"comm_ring",
"fin_succ_equiv",
"ideal",
"is_integral",
"is_integral_of_surjective",
"is_integral_quotient_map_iff",
"le_rfl",
"mv_polynomial",
"mv_polynomial.C",
"polynomial.C",
"ring_hom.comp_assoc",
"ring_hom.is_integral_of_surjective",
"ring_hom.is_integral_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_C_integral_of_surjective_of_jacobson
{R : Type*} [comm_ring R] [is_jacobson R]
{σ : Type*} [finite σ] {S : Type*} [field S] (f : mv_polynomial σ R →+* S)
(hf : function.surjective f) : (f.comp C).is_integral | begin
casesI nonempty_fintype σ,
have e := (fintype.equiv_fin σ).symm,
let f' : mv_polynomial (fin _) R →+* S :=
f.comp (rename_equiv R e).to_ring_equiv.to_ring_hom,
have hf' : function.surjective f' :=
((function.surjective.comp hf (rename_equiv R e).surjective)),
have : (f'.comp C).is_integral,
{ ... | lemma | ideal.mv_polynomial.comp_C_integral_of_surjective_of_jacobson | ring_theory | src/ring_theory/jacobson.lean | [
"ring_theory.localization.away.basic",
"ring_theory.ideal.over",
"ring_theory.jacobson_ideal"
] | [
"comm_ring",
"field",
"finite",
"fintype.equiv_fin",
"function.surjective.of_comp",
"ideal.quotient.lift",
"is_integral",
"mv_polynomial",
"nonempty_fintype",
"ring_hom.comp_assoc",
"ring_hom.ext",
"ring_hom.is_integral_trans",
"ring_hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson (I : ideal R) : ideal R | Inf {J : ideal R | I ≤ J ∧ is_maximal J} | def | ideal.jacobson | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal"
] | The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_jacobson : I ≤ jacobson I | λ x hx, mem_Inf.mpr (λ J hJ, hJ.left hx) | lemma | ideal.le_jacobson | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_idem : jacobson (jacobson I) = jacobson I | le_antisymm (Inf_le_Inf (λ J hJ, ⟨Inf_le hJ, hJ.2⟩)) le_jacobson | lemma | ideal.jacobson_idem | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_le_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_top : jacobson (⊤ : ideal R) = ⊤ | eq_top_iff.2 le_jacobson | lemma | ideal.jacobson_top | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_eq_top_iff : jacobson I = ⊤ ↔ I = ⊤ | ⟨λ H, classical.by_contradiction $ λ hi, let ⟨M, hm, him⟩ := exists_le_maximal I hi in
lt_top_iff_ne_top.1
(lt_of_le_of_lt (show jacobson I ≤ M, from Inf_le ⟨him, hm⟩) $
lt_top_iff_ne_top.2 hm.ne_top) H,
λ H, eq_top_iff.2 $ le_Inf $ λ J ⟨hij, hj⟩, H ▸ hij⟩ | theorem | ideal.jacobson_eq_top_iff | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_le",
"le_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_eq_bot : jacobson I = ⊥ → I = ⊥ | λ h, eq_bot_iff.mpr (h ▸ le_jacobson) | lemma | ideal.jacobson_eq_bot | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_eq_self_of_is_maximal [H : is_maximal I] : I.jacobson = I | le_antisymm (Inf_le ⟨le_of_eq rfl, H⟩) le_jacobson | lemma | ideal.jacobson_eq_self_of_is_maximal | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson.is_maximal [H : is_maximal I] : is_maximal (jacobson I) | ⟨⟨λ htop, H.1.1 (jacobson_eq_top_iff.1 htop),
λ J hJ, H.1.2 _ (lt_of_le_of_lt le_jacobson hJ)⟩⟩ | instance | ideal.jacobson.is_maximal | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x + z - 1 ∈ I | ⟨λ hx y, classical.by_cases
(assume hxy : I ⊔ span {y * x + 1} = ⊤,
let ⟨p, hpi, q, hq, hpq⟩ := submodule.mem_sup.1 ((eq_top_iff_one _).1 hxy) in
let ⟨r, hr⟩ := mem_span_singleton'.1 hq in
⟨r, by rw [mul_assoc, ←mul_add_one, hr, ← hpq, ← neg_sub, add_sub_cancel]; exact I.neg_mem hpi⟩)
(assume hxy : I ⊔ ... | theorem | ideal.mem_jacobson_iff | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"le_sup_left",
"mul_assoc",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mul_sub_mem_of_sub_one_mem_jacobson {I : ideal R} (r : R)
(h : r - 1 ∈ jacobson I) : ∃ s, s * r - 1 ∈ I | begin
cases mem_jacobson_iff.1 h 1 with s hs,
use s,
simpa [mul_sub] using hs
end | lemma | ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_jacobson_iff_Inf_maximal :
I.jacobson = I ↔ ∃ M : set (ideal R), (∀ J ∈ M, is_maximal J ∨ J = ⊤) ∧ I = Inf M | begin
use λ hI, ⟨{J : ideal R | I ≤ J ∧ J.is_maximal}, ⟨λ _ hJ, or.inl hJ.right, hI.symm⟩⟩,
rintros ⟨M, hM, hInf⟩,
refine le_antisymm (λ x hx, _) le_jacobson,
rw [hInf, mem_Inf],
intros I hI,
cases hM I hI with is_max is_top,
{ exact (mem_Inf.1 hx) ⟨le_Inf_iff.1 (le_of_eq hInf) I hI, is_max⟩ },
{ exact ... | theorem | ideal.eq_jacobson_iff_Inf_maximal | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal",
"is_max",
"is_top",
"submodule.mem_top"
] | An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_jacobson_iff_Inf_maximal' :
I.jacobson = I ↔ ∃ M : set (ideal R), (∀ (J ∈ M) (K : ideal R), J < K → K = ⊤) ∧ I = Inf M | eq_jacobson_iff_Inf_maximal.trans
⟨λ h, let ⟨M, hM⟩ := h in ⟨M, ⟨λ J hJ K hK, or.rec_on (hM.1 J hJ) (λ h, h.1.2 K hK)
(λ h, eq_top_iff.2 (le_of_lt (h ▸ hK))), hM.2⟩⟩,
λ h, let ⟨M, hM⟩ := h in ⟨M, ⟨λ J hJ, or.rec_on (classical.em (J = ⊤)) (λ h, or.inr h)
(λ h, or.inl ⟨⟨h, hM.1 J hJ⟩⟩), hM.2⟩⟩⟩ | theorem | ideal.eq_jacobson_iff_Inf_maximal' | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_jacobson_iff_not_mem :
I.jacobson = I ↔ ∀ x ∉ I, ∃ M : ideal R, (I ≤ M ∧ M.is_maximal) ∧ x ∉ M | begin
split,
{ intros h x hx,
erw [← h, mem_Inf] at hx,
push_neg at hx,
exact hx },
{ refine λ h, le_antisymm (λ x hx, _) le_jacobson,
contrapose hx,
erw mem_Inf,
push_neg,
exact h x hx }
end | lemma | ideal.eq_jacobson_iff_not_mem | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal"
] | An ideal `I` equals its Jacobson radical if and only if every element outside `I`
also lies outside of a maximal ideal containing `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_jacobson_of_surjective {f : R →+* S} (hf : function.surjective f) :
ring_hom.ker f ≤ I → map f (I.jacobson) = (map f I).jacobson | begin
intro h,
unfold ideal.jacobson,
have : ∀ J ∈ {J : ideal R | I ≤ J ∧ J.is_maximal}, f.ker ≤ J := λ J hJ, le_trans h hJ.left,
refine trans (map_Inf hf this) (le_antisymm _ _),
{ refine Inf_le_Inf (λ J hJ, ⟨comap f J, ⟨⟨le_comap_of_map_le hJ.1, _⟩,
map_comap_of_surjective f hf J⟩⟩),
haveI : J.is_ma... | theorem | ideal.map_jacobson_of_surjective | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_le_Inf",
"Inf_le_Inf_of_subset_insert_top",
"ideal",
"ideal.jacobson",
"ring_hom.ker",
"set.mem_insert",
"set.mem_insert_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_jacobson_of_bijective {f : R →+* S} (hf : function.bijective f) :
map f (I.jacobson) = (map f I).jacobson | map_jacobson_of_surjective hf.right
(le_trans (le_of_eq (f.injective_iff_ker_eq_bot.1 hf.left)) bot_le) | lemma | ideal.map_jacobson_of_bijective | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"bot_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_jacobson {f : R →+* S} {K : ideal S} :
comap f (K.jacobson) = Inf (comap f '' {J : ideal S | K ≤ J ∧ J.is_maximal}) | trans (comap_Inf' f _) (Inf_eq_infi).symm | lemma | ideal.comap_jacobson | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_eq_infi",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_jacobson_of_surjective {f : R →+* S} (hf : function.surjective f) {K : ideal S} :
comap f (K.jacobson) = (comap f K).jacobson | begin
unfold ideal.jacobson,
refine le_antisymm _ _,
{ refine le_trans (comap_mono (le_of_eq (trans top_inf_eq.symm Inf_insert.symm))) _,
rw [comap_Inf', Inf_eq_infi],
refine infi_le_infi_of_subset (λ J hJ, _),
have : comap f (map f J) = J := trans (comap_map_of_surjective f hf J)
(le_antisymm (... | theorem | ideal.comap_jacobson_of_surjective | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_eq_infi",
"Inf_le",
"bot_le",
"ideal",
"ideal.jacobson",
"infi_le_infi_of_subset",
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_mono {I J : ideal R} : I ≤ J → I.jacobson ≤ J.jacobson | begin
intros h x hx,
erw mem_Inf at ⊢ hx,
exact λ K ⟨hK, hK_max⟩, hx ⟨trans h hK, hK_max⟩
end | lemma | ideal.jacobson_mono | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_le_jacobson : radical I ≤ jacobson I | le_Inf (λ J hJ, (radical_eq_Inf I).symm ▸ Inf_le ⟨hJ.left, is_maximal.is_prime hJ.right⟩) | lemma | ideal.radical_le_jacobson | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_le",
"le_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical_of_eq_jacobson (h : jacobson I = I) : I.is_radical | radical_le_jacobson.trans h.le | lemma | ideal.is_radical_of_eq_jacobson | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_sub_one_mem_jacobson_bot (r : R)
(h : r - 1 ∈ jacobson (⊥ : ideal R)) : is_unit r | begin
cases exists_mul_sub_mem_of_sub_one_mem_jacobson r h with s hs,
rw [mem_bot, sub_eq_zero, mul_comm] at hs,
exact is_unit_of_mul_eq_one _ _ hs
end | lemma | ideal.is_unit_of_sub_one_mem_jacobson_bot | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal",
"is_unit",
"is_unit_of_mul_eq_one",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_jacobson_bot {x : R} : x ∈ jacobson (⊥ : ideal R) ↔ ∀ y, is_unit (x * y + 1) | ⟨λ hx y, let ⟨z, hz⟩ := (mem_jacobson_iff.1 hx) y in
is_unit_iff_exists_inv.2 ⟨z, by rwa [add_mul, one_mul, ← sub_eq_zero, mul_right_comm,
mul_comm _ z, mul_right_comm]⟩,
λ h, mem_jacobson_iff.mpr (λ y, (let ⟨b, hb⟩ := is_unit_iff_exists_inv.1 (h y) in
⟨b, (submodule.mem_bot R).2 (hb ▸ (by ring))⟩))⟩ | lemma | ideal.mem_jacobson_bot | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal",
"is_unit",
"mul_comm",
"mul_right_comm",
"one_mul",
"ring",
"submodule.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_eq_iff_jacobson_quotient_eq_bot :
I.jacobson = I ↔ jacobson (⊥ : ideal (R ⧸ I)) = ⊥ | begin
have hf : function.surjective (quotient.mk I) := submodule.quotient.mk_surjective I,
split,
{ intro h,
replace h := congr_arg (map (quotient.mk I)) h,
rw map_jacobson_of_surjective hf (le_of_eq mk_ker) at h,
simpa using h },
{ intro h,
replace h := congr_arg (comap (quotient.mk I)) h,
... | theorem | ideal.jacobson_eq_iff_jacobson_quotient_eq_bot | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal",
"submodule.quotient.mk_surjective"
] | An ideal `I` of `R` is equal to its Jacobson radical if and only if
the Jacobson radical of the quotient ring `R/I` is the zero ideal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical (⊥ : ideal (R ⧸ I)) = jacobson ⊥ | begin
have hf : function.surjective (quotient.mk I) := submodule.quotient.mk_surjective I,
split,
{ intro h,
have := congr_arg (map (quotient.mk I)) h,
rw [map_radical_of_surjective hf (le_of_eq mk_ker),
map_jacobson_of_surjective hf (le_of_eq mk_ker)] at this,
simpa using this },
{ intro h,
... | theorem | ideal.radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal",
"submodule.quotient.mk_surjective"
] | The standard radical and Jacobson radical of an ideal `I` of `R` are equal if and only if
the nilradical and Jacobson radical of the quotient ring `R/I` coincide | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
jacobson_radical_eq_jacobson :
I.radical.jacobson = I.jacobson | le_antisymm (le_trans (le_of_eq (congr_arg jacobson (radical_eq_Inf I)))
(Inf_le_Inf (λ J hJ, ⟨Inf_le ⟨hJ.1, hJ.2.is_prime⟩, hJ.2⟩))) (jacobson_mono le_radical) | lemma | ideal.jacobson_radical_eq_jacobson | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_le_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_bot_polynomial_le_Inf_map_maximal :
jacobson (⊥ : ideal R[X]) ≤ Inf (map (C : R →+* R[X]) '' {J : ideal R | J.is_maximal}) | begin
refine le_Inf (λ J, exists_imp_distrib.2 (λ j hj, _)),
haveI : j.is_maximal := hj.1,
refine trans (jacobson_mono bot_le) (le_of_eq _ : J.jacobson ≤ J),
suffices : (⊥ : ideal (polynomial (R ⧸ j))).jacobson = ⊥,
{ rw [← hj.2, jacobson_eq_iff_jacobson_quotient_eq_bot],
replace this :=
congr_arg (ma... | lemma | ideal.jacobson_bot_polynomial_le_Inf_map_maximal | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"bot_le",
"ideal",
"le_Inf",
"polynomial",
"ring_equiv.bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : ideal R) = ⊥) :
jacobson (⊥ : ideal R[X]) = ⊥ | begin
refine eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_Inf_map_maximal _),
refine (λ f hf, ((submodule.mem_bot _).2 (polynomial.ext (λ n, trans _ (coeff_zero n).symm)))),
suffices : f.coeff n ∈ ideal.jacobson ⊥, by rwa [h, submodule.mem_bot] at this,
exact mem_Inf.2 (λ j hj, (mem_map_C_iff.1 ((mem_Inf.1... | lemma | ideal.jacobson_bot_polynomial_of_jacobson_bot | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal",
"ideal.jacobson",
"polynomial.ext",
"submodule.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local (I : ideal R) : Prop | (out : is_maximal (jacobson I)) | class | ideal.is_local | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal"
] | An ideal `I` is local iff its Jacobson radical is maximal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_iff {I : ideal R} : is_local I ↔ is_maximal (jacobson I) | ⟨λ h, h.1, λ h, ⟨h⟩⟩ | theorem | ideal.is_local_iff | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_of_is_maximal_radical {I : ideal R} (hi : is_maximal (radical I)) : is_local I | ⟨have radical I = jacobson I,
from le_antisymm (le_Inf $ λ M ⟨him, hm⟩, hm.is_prime.radical_le_iff.2 him)
(Inf_le ⟨le_radical, hi⟩),
show is_maximal (jacobson I), from this ▸ hi⟩ | theorem | ideal.is_local_of_is_maximal_radical | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_le",
"ideal",
"le_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local.le_jacobson {I J : ideal R} (hi : is_local I) (hij : I ≤ J) (hj : J ≠ ⊤) :
J ≤ jacobson I | let ⟨M, hm, hjm⟩ := exists_le_maximal J hj in
le_trans hjm $ le_of_eq $ eq.symm $ hi.1.eq_of_le hm.1.1 $ Inf_le ⟨le_trans hij hjm, hm⟩ | theorem | ideal.is_local.le_jacobson | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_le",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local.mem_jacobson_or_exists_inv {I : ideal R} (hi : is_local I) (x : R) :
x ∈ jacobson I ∨ ∃ y, y * x - 1 ∈ I | classical.by_cases
(assume h : I ⊔ span {x} = ⊤,
let ⟨p, hpi, q, hq, hpq⟩ := submodule.mem_sup.1 ((eq_top_iff_one _).1 h) in
let ⟨r, hr⟩ := mem_span_singleton.1 hq in
or.inr ⟨r, by rw [← hpq, mul_comm, ← hr, ← neg_sub, add_sub_cancel]; exact I.neg_mem hpi⟩)
(assume h : I ⊔ span {x} ≠ ⊤,
or.inl $ le_... | theorem | ideal.is_local.mem_jacobson_or_exists_inv | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"dvd_refl",
"ideal",
"le_sup_left",
"le_sup_right",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_primary_of_is_maximal_radical [comm_ring R] {I : ideal R} (hi : is_maximal (radical I)) :
is_primary I | have radical I = jacobson I,
from le_antisymm (le_Inf $ λ M ⟨him, hm⟩, hm.is_prime.radical_le_iff.2 him)
(Inf_le ⟨le_radical, hi⟩),
⟨ne_top_of_lt $ lt_of_le_of_lt le_radical (lt_top_iff_ne_top.2 hi.1.1),
λ x y hxy, ((is_local_of_is_maximal_radical hi).mem_jacobson_or_exists_inv y).symm.imp
(λ ⟨z, hz⟩, by rw [← mul_... | theorem | ideal.is_primary_of_is_maximal_radical | ring_theory | src/ring_theory/jacobson_ideal.lean | [
"ring_theory.ideal.quotient",
"ring_theory.polynomial.quotient"
] | [
"Inf_le",
"comm_ring",
"ideal",
"le_Inf",
"mul_left_comm",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.ideal : ideal (S ⊗[R] S) | ring_hom.ker (tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S) | abbreviation | kaehler_differential.ideal | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"ideal",
"ring_hom.ker"
] | The kernel of the multiplication map `S ⊗[R] S →ₐ[R] S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.one_smul_sub_smul_one_mem_ideal (a : S) :
(1 : S) ⊗ₜ[R] a - a ⊗ₜ[R] (1 : S) ∈ kaehler_differential.ideal R S | by simp [ring_hom.mem_ker] | lemma | kaehler_differential.one_smul_sub_smul_one_mem_ideal | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.ideal",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivation.tensor_product_to (D : derivation R S M) : S ⊗[R] S →ₗ[S] M | tensor_product.algebra_tensor_module.lift ((linear_map.lsmul S (S →ₗ[R] M)).flip D.to_linear_map) | def | derivation.tensor_product_to | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"linear_map.lsmul",
"tensor_product.algebra_tensor_module.lift"
] | For a `R`-derivation `S → M`, this is the map `S ⊗[R] S →ₗ[S] M` sending `s ⊗ₜ t ↦ s • D t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
derivation.tensor_product_to_tmul (D : derivation R S M) (s t : S) :
D.tensor_product_to (s ⊗ₜ t) = s • D t | rfl | lemma | derivation.tensor_product_to_tmul | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivation.tensor_product_to_mul (D : derivation R S M) (x y : S ⊗[R] S) :
D.tensor_product_to (x * y) = tensor_product.lmul' R x • D.tensor_product_to y +
tensor_product.lmul' R y • D.tensor_product_to x | begin
apply tensor_product.induction_on x,
{ rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero] },
swap, { rintros, simp only [add_mul, map_add, add_smul, *, smul_add], rw add_add_add_comm },
intros x₁ x₂,
apply tensor_product.induction_on y,
{ rw [mul_zero, map_zero, map_zero, zero_smul, smu... | lemma | derivation.tensor_product_to_mul | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"add_smul",
"derivation",
"derivation.tensor_product_to",
"mul_assoc",
"mul_comm",
"mul_right_comm",
"mul_zero",
"smul_add",
"smul_smul",
"smul_zero",
"tensor_product.induction_on",
"tensor_product.lift.tmul'",
"zero_mul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.submodule_span_range_eq_ideal :
submodule.span S (set.range $ λ s : S, (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
(kaehler_differential.ideal R S).restrict_scalars S | begin
apply le_antisymm,
{ rw submodule.span_le,
rintros _ ⟨s, rfl⟩,
exact kaehler_differential.one_smul_sub_smul_one_mem_ideal _ _ },
{ rintros x (hx : _ = _),
have : x - (tensor_product.lmul' R x) ⊗ₜ[R] (1 : S) = x,
{ rw [hx, tensor_product.zero_tmul, sub_zero] },
rw ← this,
clear this... | lemma | kaehler_differential.submodule_span_range_eq_ideal | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.ideal",
"kaehler_differential.one_smul_sub_smul_one_mem_ideal",
"mul_one",
"restrict_scalars",
"set.mem_range_self",
"set.range",
"smul_eq_mul",
"smul_sub",
"submodule.smul_mem",
"submodule.span",
"submodule.span_le",
"submodule.subset_span",
"tensor_product.add_tmul",
... | The kernel of `S ⊗[R] S →ₐ[R] S` is generated by `1 ⊗ s - s ⊗ 1` as a `S`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.span_range_eq_ideal :
ideal.span (set.range $ λ s : S, (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
kaehler_differential.ideal R S | begin
apply le_antisymm,
{ rw ideal.span_le,
rintros _ ⟨s, rfl⟩,
exact kaehler_differential.one_smul_sub_smul_one_mem_ideal _ _ },
{ change (kaehler_differential.ideal R S).restrict_scalars S ≤ (ideal.span _).restrict_scalars S,
rw [← kaehler_differential.submodule_span_range_eq_ideal, ideal.span],
... | lemma | kaehler_differential.span_range_eq_ideal | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"ideal.span",
"ideal.span_le",
"kaehler_differential.ideal",
"kaehler_differential.one_smul_sub_smul_one_mem_ideal",
"kaehler_differential.submodule_span_range_eq_ideal",
"restrict_scalars",
"set.range",
"submodule.span_span_of_tower",
"submodule.subset_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential : Type* | (kaehler_differential.ideal R S).cotangent | def | kaehler_differential | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.ideal"
] | The module of Kähler differentials (Kahler differentials, Kaehler differentials).
This is implemented as `I / I ^ 2` with `I` the kernel of the multiplication map `S ⊗[R] S →ₐ[R] S`.
To view elements as a linear combination of the form `s • D s'`, use
`kaehler_differential.tensor_product_to_surjective` and `derivation.... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.module' {R' : Type*} [comm_ring R'] [algebra R' S]
[smul_comm_class R R' S] :
module R' Ω[S⁄R] | submodule.quotient.module' _ | instance | kaehler_differential.module' | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"algebra",
"comm_ring",
"module",
"smul_comm_class",
"submodule.quotient.module'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.is_scalar_tower_of_tower {R₁ R₂ : Type*} [comm_ring R₁] [comm_ring R₂]
[algebra R₁ S] [algebra R₂ S] [has_smul R₁ R₂]
[smul_comm_class R R₁ S] [smul_comm_class R R₂ S] [is_scalar_tower R₁ R₂ S] :
is_scalar_tower R₁ R₂ Ω[S⁄R] | submodule.quotient.is_scalar_tower _ _ | instance | kaehler_differential.is_scalar_tower_of_tower | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"algebra",
"comm_ring",
"has_smul",
"is_scalar_tower",
"smul_comm_class",
"submodule.quotient.is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.is_scalar_tower' :
is_scalar_tower R (S ⊗[R] S) Ω[S⁄R] | submodule.quotient.is_scalar_tower _ _ | instance | kaehler_differential.is_scalar_tower' | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"is_scalar_tower",
"submodule.quotient.is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.from_ideal : kaehler_differential.ideal R S →ₗ[S ⊗[R] S] Ω[S⁄R] | (kaehler_differential.ideal R S).to_cotangent | def | kaehler_differential.from_ideal | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.ideal"
] | The quotient map `I → Ω[S⁄R]` with `I` being the kernel of `S ⊗[R] S → S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.D_linear_map : S →ₗ[R] Ω[S⁄R] | ((kaehler_differential.from_ideal R S).restrict_scalars R).comp
((tensor_product.include_right.to_linear_map - tensor_product.include_left.to_linear_map :
S →ₗ[R] S ⊗[R] S).cod_restrict ((kaehler_differential.ideal R S).restrict_scalars R)
(kaehler_differential.one_smul_sub_smul_one_mem_ideal R) : _ →ₗ[R] _... | def | kaehler_differential.D_linear_map | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.from_ideal",
"kaehler_differential.ideal",
"kaehler_differential.one_smul_sub_smul_one_mem_ideal",
"restrict_scalars"
] | (Implementation) The underlying linear map of the derivation into `Ω[S⁄R]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.D_linear_map_apply (s : S) :
kaehler_differential.D_linear_map R S s = (kaehler_differential.ideal R S).to_cotangent
⟨1 ⊗ₜ s - s ⊗ₜ 1, kaehler_differential.one_smul_sub_smul_one_mem_ideal R s⟩ | rfl | lemma | kaehler_differential.D_linear_map_apply | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.D_linear_map",
"kaehler_differential.ideal",
"kaehler_differential.one_smul_sub_smul_one_mem_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.D : derivation R S Ω[S⁄R] | { map_one_eq_zero' := begin
dsimp only [kaehler_differential.D_linear_map_apply],
rw [ideal.to_cotangent_eq_zero, subtype.coe_mk, sub_self],
exact zero_mem _
end,
leibniz' := λ a b, begin
dsimp only [kaehler_differential.D_linear_map_apply],
rw [← linear_map.map_smul_of_tower ((kaehler_different... | def | kaehler_differential.D | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"ideal.to_cotangent_eq",
"ideal.to_cotangent_eq_zero",
"kaehler_differential.D_linear_map",
"kaehler_differential.D_linear_map_apply",
"kaehler_differential.ideal",
"kaehler_differential.one_smul_sub_smul_one_mem_ideal",
"linear_map.map_smul_of_tower",
"mul_comm",
"mul_one",
"pow_t... | The universal derivation into `Ω[S⁄R]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.D_apply (s : S) :
kaehler_differential.D R S s = (kaehler_differential.ideal R S).to_cotangent
⟨1 ⊗ₜ s - s ⊗ₜ 1, kaehler_differential.one_smul_sub_smul_one_mem_ideal R s⟩ | rfl | lemma | kaehler_differential.D_apply | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.D",
"kaehler_differential.ideal",
"kaehler_differential.one_smul_sub_smul_one_mem_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.span_range_derivation :
submodule.span S (set.range $ kaehler_differential.D R S) = ⊤ | begin
rw _root_.eq_top_iff,
rintros x -,
obtain ⟨⟨x, hx⟩, rfl⟩ := ideal.to_cotangent_surjective _ x,
have : x ∈ (kaehler_differential.ideal R S).restrict_scalars S := hx,
rw ← kaehler_differential.submodule_span_range_eq_ideal at this,
suffices : ∃ hx, (kaehler_differential.ideal R S).to_cotangent ⟨x, hx⟩ ∈... | lemma | kaehler_differential.span_range_derivation | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"ideal.to_cotangent_surjective",
"kaehler_differential.D",
"kaehler_differential.D_linear_map_apply",
"kaehler_differential.ideal",
"kaehler_differential.submodule_span_range_eq_ideal",
"restrict_scalars",
"set.range",
"submodule.add_mem",
"submodule.smul_mem",
"submodule.span",
"submodule.span_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivation.lift_kaehler_differential (D : derivation R S M) : Ω[S⁄R] →ₗ[S] M | begin
refine ((kaehler_differential.ideal R S • ⊤ :
submodule (S ⊗[R] S) (kaehler_differential.ideal R S)).restrict_scalars S).liftq _ _,
{ exact D.tensor_product_to.comp ((kaehler_differential.ideal R S).subtype.restrict_scalars S) },
{ intros x hx,
change _ = _,
apply submodule.smul_induction_on hx;... | def | derivation.lift_kaehler_differential | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"derivation.tensor_product_to_mul",
"kaehler_differential.ideal",
"restrict_scalars",
"submodule",
"submodule.smul_induction_on",
"zero_smul"
] | The linear map from `Ω[S⁄R]`, associated with a derivation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
derivation.lift_kaehler_differential_apply (D : derivation R S M) (x) :
D.lift_kaehler_differential
((kaehler_differential.ideal R S).to_cotangent x) = D.tensor_product_to x | rfl | lemma | derivation.lift_kaehler_differential_apply | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"kaehler_differential.ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivation.lift_kaehler_differential_comp (D : derivation R S M) :
D.lift_kaehler_differential.comp_der (kaehler_differential.D R S) = D | begin
ext a,
dsimp [kaehler_differential.D_apply],
refine (D.lift_kaehler_differential_apply _).trans _,
rw [subtype.coe_mk, map_sub, derivation.tensor_product_to_tmul,
derivation.tensor_product_to_tmul, one_smul, D.map_one_eq_zero, smul_zero, sub_zero],
end | lemma | derivation.lift_kaehler_differential_comp | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"derivation.tensor_product_to_tmul",
"kaehler_differential.D",
"kaehler_differential.D_apply",
"one_smul",
"smul_zero",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivation.lift_kaehler_differential_comp_D (D' : derivation R S M) (x : S) :
D'.lift_kaehler_differential (kaehler_differential.D R S x) = D' x | derivation.congr_fun D'.lift_kaehler_differential_comp x | lemma | derivation.lift_kaehler_differential_comp_D | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"derivation.congr_fun",
"kaehler_differential.D"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivation.lift_kaehler_differential_unique
(f f' : Ω[S⁄R] →ₗ[S] M)
(hf : f.comp_der (kaehler_differential.D R S) =
f'.comp_der (kaehler_differential.D R S)) :
f = f' | begin
apply linear_map.ext,
intro x,
have : x ∈ submodule.span S (set.range $ kaehler_differential.D R S),
{ rw kaehler_differential.span_range_derivation, trivial },
apply submodule.span_induction this,
{ rintros _ ⟨x, rfl⟩, exact congr_arg (λ D : derivation R S M, D x) hf },
{ rw [map_zero, map_zero] },... | lemma | derivation.lift_kaehler_differential_unique | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"kaehler_differential.D",
"kaehler_differential.span_range_derivation",
"linear_map.ext",
"set.range",
"submodule.span",
"submodule.span_induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivation.lift_kaehler_differential_D :
(kaehler_differential.D R S).lift_kaehler_differential = linear_map.id | derivation.lift_kaehler_differential_unique _ _
(kaehler_differential.D R S).lift_kaehler_differential_comp | lemma | derivation.lift_kaehler_differential_D | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation.lift_kaehler_differential_unique",
"kaehler_differential.D",
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.D_tensor_product_to (x : kaehler_differential.ideal R S) :
(kaehler_differential.D R S).tensor_product_to x =
(kaehler_differential.ideal R S).to_cotangent x | begin
rw [← derivation.lift_kaehler_differential_apply, derivation.lift_kaehler_differential_D],
refl,
end | lemma | kaehler_differential.D_tensor_product_to | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation.lift_kaehler_differential_D",
"derivation.lift_kaehler_differential_apply",
"kaehler_differential.D",
"kaehler_differential.ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.tensor_product_to_surjective :
function.surjective (kaehler_differential.D R S).tensor_product_to | begin
intro x, obtain ⟨x, rfl⟩ := (kaehler_differential.ideal R S).to_cotangent_surjective x,
exact ⟨x, kaehler_differential.D_tensor_product_to x⟩
end | lemma | kaehler_differential.tensor_product_to_surjective | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.D",
"kaehler_differential.D_tensor_product_to",
"kaehler_differential.ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.linear_map_equiv_derivation : (Ω[S⁄R] →ₗ[S] M) ≃ₗ[S] derivation R S M | { inv_fun := derivation.lift_kaehler_differential,
left_inv := λ f, derivation.lift_kaehler_differential_unique _ _
(derivation.lift_kaehler_differential_comp _),
right_inv := derivation.lift_kaehler_differential_comp,
..(derivation.llcomp.flip $ kaehler_differential.D R S) } | def | kaehler_differential.linear_map_equiv_derivation | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"derivation.lift_kaehler_differential",
"derivation.lift_kaehler_differential_comp",
"derivation.lift_kaehler_differential_unique",
"inv_fun",
"kaehler_differential.D"
] | The `S`-linear maps from `Ω[S⁄R]` to `M` are (`S`-linearly) equivalent to `R`-derivations
from `S` to `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.quotient_cotangent_ideal_ring_equiv :
(S ⊗ S ⧸ kaehler_differential.ideal R S ^ 2) ⧸
(kaehler_differential.ideal R S).cotangent_ideal ≃+* S | begin
have : function.right_inverse tensor_product.include_left
(↑(tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S) : S ⊗[R] S →+* S),
{ intro x, rw [alg_hom.coe_to_ring_hom, ← alg_hom.comp_apply,
tensor_product.lmul'_comp_include_left], refl },
refine (ideal.quot_cotangent _).trans _,
refine (ideal.quot_eq... | def | kaehler_differential.quotient_cotangent_ideal_ring_equiv | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"alg_hom.coe_to_ring_hom",
"alg_hom.comp_apply",
"ideal.quot_cotangent",
"ideal.quot_equiv_of_eq",
"kaehler_differential.ideal",
"ring_hom.quotient_ker_equiv_of_right_inverse"
] | The quotient ring of `S ⊗ S ⧸ J ^ 2` by `Ω[S⁄R]` is isomorphic to `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.quotient_cotangent_ideal :
((S ⊗ S ⧸ kaehler_differential.ideal R S ^ 2) ⧸
(kaehler_differential.ideal R S).cotangent_ideal) ≃ₐ[S] S | { commutes' := (kaehler_differential.quotient_cotangent_ideal_ring_equiv R S).apply_symm_apply,
..kaehler_differential.quotient_cotangent_ideal_ring_equiv R S } | def | kaehler_differential.quotient_cotangent_ideal | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.ideal",
"kaehler_differential.quotient_cotangent_ideal_ring_equiv"
] | The quotient ring of `S ⊗ S ⧸ J ^ 2` by `Ω[S⁄R]` is isomorphic to `S` as an `S`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.End_equiv_aux (f : S →ₐ[R] S ⊗ S ⧸ kaehler_differential.ideal R S ^ 2) :
(ideal.quotient.mkₐ R (kaehler_differential.ideal R S).cotangent_ideal).comp f =
is_scalar_tower.to_alg_hom R S _ ↔
(tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S).ker_square_lift.comp f = alg_hom.id R S | begin
rw [alg_hom.ext_iff, alg_hom.ext_iff],
apply forall_congr,
intro x,
have e₁ : (tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S).ker_square_lift (f x) =
kaehler_differential.quotient_cotangent_ideal_ring_equiv R S
(ideal.quotient.mk (kaehler_differential.ideal R S).cotangent_ideal $ f x),
{ generaliz... | lemma | kaehler_differential.End_equiv_aux | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"alg_hom.ext_iff",
"alg_hom.id",
"ideal.quotient.mk",
"ideal.quotient.mk_surjective",
"ideal.quotient.mkₐ",
"is_scalar_tower.to_alg_hom",
"kaehler_differential.ideal",
"kaehler_differential.quotient_cotangent_ideal_ring_equiv",
"mul_one",
"ring_equiv.congr_arg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.End_equiv_derivation' :
derivation R S Ω[S⁄R] ≃ₗ[R] derivation R S _ | linear_equiv.comp_der ((kaehler_differential.ideal R S).cotangent_equiv_ideal.restrict_scalars S) | def | kaehler_differential.End_equiv_derivation' | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"kaehler_differential.ideal",
"linear_equiv.comp_der"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.End_equiv_aux_equiv :
{f // (ideal.quotient.mkₐ R (kaehler_differential.ideal R S).cotangent_ideal).comp f =
is_scalar_tower.to_alg_hom R S _ } ≃
{ f // (tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S).ker_square_lift.comp f = alg_hom.id R S } | (equiv.refl _).subtype_equiv (kaehler_differential.End_equiv_aux R S) | def | kaehler_differential.End_equiv_aux_equiv | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"alg_hom.id",
"equiv.refl",
"ideal.quotient.mkₐ",
"is_scalar_tower.to_alg_hom",
"kaehler_differential.End_equiv_aux",
"kaehler_differential.ideal"
] | (Implementation) An `equiv` version of `kaehler_differential.End_equiv_aux`.
Used in `kaehler_differential.End_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.End_equiv :
module.End S Ω[S⁄R] ≃
{ f // (tensor_product.lmul' R : S ⊗[R] S →ₐ[R] S).ker_square_lift.comp f = alg_hom.id R S } | (kaehler_differential.linear_map_equiv_derivation R S).to_equiv.trans $
(kaehler_differential.End_equiv_derivation' R S).to_equiv.trans $
(derivation_to_square_zero_equiv_lift
(kaehler_differential.ideal R S).cotangent_ideal
(kaehler_differential.ideal R S).cotangent_ideal_square).trans $
kaehler_differential... | def | kaehler_differential.End_equiv | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"alg_hom.id",
"derivation_to_square_zero_equiv_lift",
"kaehler_differential.End_equiv_aux_equiv",
"kaehler_differential.End_equiv_derivation'",
"kaehler_differential.ideal",
"kaehler_differential.linear_map_equiv_derivation",
"module.End"
] | The endomorphisms of `Ω[S⁄R]` corresponds to sections of the surjection `S ⊗[R] S ⧸ J ^ 2 →ₐ[R] S`,
with `J` being the kernel of the multiplication map `S ⊗[R] S →ₐ[R] S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.ker_total : submodule S (S →₀ S) | submodule.span S
((set.range (λ (x : S × S), single x.1 1 + single x.2 1 - single (x.1 + x.2) 1)) ∪
(set.range (λ (x : S × S), single x.2 x.1 + single x.1 x.2 - single (x.1 * x.2) 1)) ∪
(set.range (λ x : R, single (algebra_map R S x) 1))) | def | kaehler_differential.ker_total | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"algebra_map",
"set.range",
"submodule",
"submodule.span"
] | The `S`-submodule of `S →₀ S` (the direct sum of copies of `S` indexed by `S`) generated by
the relations:
1. `dx + dy = d(x + y)`
2. `x dy + y dx = d(x * y)`
3. `dr = 0` for `r ∈ R`
where `db` is the unit in the copy of `S` with index `b`.
This is the kernel of the surjection `finsupp.total S Ω[S⁄R] S (kaehler_differ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.ker_total_mkq_single_add (x y z) :
(z 𝖣 (x + y)) = (z 𝖣 x) + (z 𝖣 y) | begin
rw [← map_add, eq_comm, ← sub_eq_zero, ← map_sub, submodule.mkq_apply,
submodule.quotient.mk_eq_zero],
simp_rw [← finsupp.smul_single_one _ z, ← smul_add, ← smul_sub],
exact submodule.smul_mem _ _ (submodule.subset_span (or.inl $ or.inl $ ⟨⟨_, _⟩, rfl⟩)),
end | lemma | kaehler_differential.ker_total_mkq_single_add | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"finsupp.smul_single_one",
"smul_add",
"smul_sub",
"submodule.mkq_apply",
"submodule.quotient.mk_eq_zero",
"submodule.smul_mem",
"submodule.subset_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.ker_total_mkq_single_mul (x y z) :
(z 𝖣 (x * y)) = ((z * x) 𝖣 y) + ((z * y) 𝖣 x) | begin
rw [← map_add, eq_comm, ← sub_eq_zero, ← map_sub, submodule.mkq_apply,
submodule.quotient.mk_eq_zero],
simp_rw [← finsupp.smul_single_one _ z, ← @smul_eq_mul _ _ z,
← finsupp.smul_single, ← smul_add, ← smul_sub],
exact submodule.smul_mem _ _ (submodule.subset_span (or.inl $ or.inr $ ⟨⟨_, _⟩, rfl⟩)),... | lemma | kaehler_differential.ker_total_mkq_single_mul | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"finsupp.smul_single",
"finsupp.smul_single_one",
"smul_add",
"smul_eq_mul",
"smul_sub",
"submodule.mkq_apply",
"submodule.quotient.mk_eq_zero",
"submodule.smul_mem",
"submodule.subset_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.ker_total_mkq_single_algebra_map (x y) :
(y 𝖣 (algebra_map R S x)) = 0 | begin
rw [submodule.mkq_apply, submodule.quotient.mk_eq_zero, ← finsupp.smul_single_one _ y],
exact submodule.smul_mem _ _ (submodule.subset_span (or.inr $ ⟨_, rfl⟩)),
end | lemma | kaehler_differential.ker_total_mkq_single_algebra_map | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"algebra_map",
"finsupp.smul_single_one",
"submodule.mkq_apply",
"submodule.quotient.mk_eq_zero",
"submodule.smul_mem",
"submodule.subset_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.ker_total_mkq_single_algebra_map_one (x) :
(x 𝖣 1) = 0 | begin
rw [← (algebra_map R S).map_one, kaehler_differential.ker_total_mkq_single_algebra_map],
end | lemma | kaehler_differential.ker_total_mkq_single_algebra_map_one | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"algebra_map",
"kaehler_differential.ker_total_mkq_single_algebra_map",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.ker_total_mkq_single_smul (r : R) (x y) :
(y 𝖣 (r • x)) = r • (y 𝖣 x) | begin
rw [algebra.smul_def, kaehler_differential.ker_total_mkq_single_mul,
kaehler_differential.ker_total_mkq_single_algebra_map, add_zero,
← linear_map.map_smul_of_tower, finsupp.smul_single, mul_comm, algebra.smul_def],
end | lemma | kaehler_differential.ker_total_mkq_single_smul | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"algebra.smul_def",
"finsupp.smul_single",
"kaehler_differential.ker_total_mkq_single_algebra_map",
"kaehler_differential.ker_total_mkq_single_mul",
"linear_map.map_smul_of_tower",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.derivation_quot_ker_total :
derivation R S ((S →₀ S) ⧸ kaehler_differential.ker_total R S) | { to_fun := λ x, 1 𝖣 x,
map_add' := λ x y, kaehler_differential.ker_total_mkq_single_add _ _ _ _ _,
map_smul' := λ r s, kaehler_differential.ker_total_mkq_single_smul _ _ _ _ _,
map_one_eq_zero' := kaehler_differential.ker_total_mkq_single_algebra_map_one _ _ _,
leibniz' := λ a b, (kaehler_differential.ker_tot... | def | kaehler_differential.derivation_quot_ker_total | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"finsupp.smul_single_one",
"kaehler_differential.ker_total",
"kaehler_differential.ker_total_mkq_single_add",
"kaehler_differential.ker_total_mkq_single_algebra_map_one",
"kaehler_differential.ker_total_mkq_single_mul",
"kaehler_differential.ker_total_mkq_single_smul"
] | The (universal) derivation into `(S →₀ S) ⧸ kaehler_differential.ker_total R S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.derivation_quot_ker_total_apply (x) :
kaehler_differential.derivation_quot_ker_total R S x = (1 𝖣 x) | rfl | lemma | kaehler_differential.derivation_quot_ker_total_apply | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.derivation_quot_ker_total"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.derivation_quot_ker_total_lift_comp_total :
(kaehler_differential.derivation_quot_ker_total R S).lift_kaehler_differential.comp
(finsupp.total S Ω[S⁄R] S (kaehler_differential.D R S)) = submodule.mkq _ | begin
apply finsupp.lhom_ext,
intros a b,
conv_rhs { rw [← finsupp.smul_single_one a b, linear_map.map_smul] },
simp [kaehler_differential.derivation_quot_ker_total_apply],
end | lemma | kaehler_differential.derivation_quot_ker_total_lift_comp_total | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"finsupp.lhom_ext",
"finsupp.smul_single_one",
"finsupp.total",
"kaehler_differential.D",
"kaehler_differential.derivation_quot_ker_total",
"kaehler_differential.derivation_quot_ker_total_apply",
"linear_map.map_smul",
"submodule.mkq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.ker_total_eq :
(finsupp.total S Ω[S⁄R] S (kaehler_differential.D R S)).ker =
kaehler_differential.ker_total R S | begin
apply le_antisymm,
{ conv_rhs { rw ← (kaehler_differential.ker_total R S).ker_mkq },
rw ← kaehler_differential.derivation_quot_ker_total_lift_comp_total,
exact linear_map.ker_le_ker_comp _ _ },
{ rw [kaehler_differential.ker_total, submodule.span_le],
rintros _ ((⟨⟨x, y⟩, rfl⟩|⟨⟨x, y⟩, rfl⟩)|⟨x,... | lemma | kaehler_differential.ker_total_eq | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"finsupp.total",
"kaehler_differential.D",
"kaehler_differential.derivation_quot_ker_total_lift_comp_total",
"kaehler_differential.ker_total",
"linear_map.ker_le_ker_comp",
"linear_map.mem_ker",
"submodule.span_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.total_surjective :
function.surjective (finsupp.total S Ω[S⁄R] S (kaehler_differential.D R S)) | begin
rw [← linear_map.range_eq_top, finsupp.range_total, kaehler_differential.span_range_derivation],
end | lemma | kaehler_differential.total_surjective | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"finsupp.range_total",
"finsupp.total",
"kaehler_differential.D",
"kaehler_differential.span_range_derivation",
"linear_map.range_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.quot_ker_total_equiv :
((S →₀ S) ⧸ kaehler_differential.ker_total R S) ≃ₗ[S] Ω[S⁄R] | { inv_fun := (kaehler_differential.derivation_quot_ker_total R S).lift_kaehler_differential,
left_inv := begin
intro x,
obtain ⟨x, rfl⟩ := submodule.mkq_surjective _ x,
exact linear_map.congr_fun
(kaehler_differential.derivation_quot_ker_total_lift_comp_total R S : _) x,
end,
right_inv := begin
... | def | kaehler_differential.quot_ker_total_equiv | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"finsupp.total",
"inv_fun",
"kaehler_differential.D",
"kaehler_differential.derivation_quot_ker_total",
"kaehler_differential.derivation_quot_ker_total_lift_comp_total",
"kaehler_differential.ker_total",
"kaehler_differential.ker_total_eq",
"kaehler_differential.total_surjective",
"linear_map.congr_... | `Ω[S⁄R]` is isomorphic to `S` copies of `S` with kernel `kaehler_differential.ker_total`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.quot_ker_total_equiv_symm_comp_D :
(kaehler_differential.quot_ker_total_equiv R S).symm.to_linear_map.comp_der
(kaehler_differential.D R S) = kaehler_differential.derivation_quot_ker_total R S | by convert
(kaehler_differential.derivation_quot_ker_total R S).lift_kaehler_differential_comp using 0 | lemma | kaehler_differential.quot_ker_total_equiv_symm_comp_D | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.D",
"kaehler_differential.derivation_quot_ker_total",
"kaehler_differential.quot_ker_total_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.ker_total_map (h : function.surjective (algebra_map A B)) :
(kaehler_differential.ker_total R A).map finsupp_map ⊔
submodule.span A (set.range (λ x : S, single (algebra_map S B x) (1 : B))) =
(kaehler_differential.ker_total S B).restrict_scalars _ | begin
rw [kaehler_differential.ker_total, submodule.map_span, kaehler_differential.ker_total,
submodule.restrict_scalars_span _ _ h],
simp_rw [set.image_union, submodule.span_union, ← set.image_univ, set.image_image,
set.image_univ, map_sub, map_add],
simp only [linear_map.comp_apply, finsupp.map_range.li... | lemma | kaehler_differential.ker_total_map | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"alg_hom.to_linear_map_apply",
"algebra.linear_map_apply",
"algebra_map",
"finsupp.lmap_domain_apply",
"finsupp.map_domain_single",
"finsupp.map_range_single",
"is_scalar_tower.algebra_map_apply",
"kaehler_differential.ker_total",
"linear_map.comp_apply",
"map_mul",
"map_one",
"restrict_scalar... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivation.comp_algebra_map [module A M] [module B M] [is_scalar_tower A B M]
(d : derivation R B M) : derivation R A M | { map_one_eq_zero' := by simp,
leibniz' := λ a b, by simp,
to_linear_map := d.to_linear_map.comp (is_scalar_tower.to_alg_hom R A B).to_linear_map } | def | derivation.comp_algebra_map | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation",
"is_scalar_tower",
"is_scalar_tower.to_alg_hom",
"module"
] | For a tower `R → A → B` and an `R`-derivation `B → M`, we may compose with `A → B` to obtain an
`R`-derivation `A → M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.map : Ω[A⁄R] →ₗ[A] Ω[B⁄S] | derivation.lift_kaehler_differential
(((kaehler_differential.D S B).restrict_scalars R).comp_algebra_map A) | def | kaehler_differential.map | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation.lift_kaehler_differential",
"kaehler_differential.D",
"restrict_scalars"
] | The map `Ω[A⁄R] →ₗ[A] Ω[B⁄R]` given a square
A --→ B
↑ ↑
| |
R --→ S | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.map_comp_der :
(kaehler_differential.map R S A B).comp_der (kaehler_differential.D R A) =
(((kaehler_differential.D S B).restrict_scalars R).comp_algebra_map A) | derivation.lift_kaehler_differential_comp _ | lemma | kaehler_differential.map_comp_der | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"derivation.lift_kaehler_differential_comp",
"kaehler_differential.D",
"kaehler_differential.map",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.map_D (x : A) :
kaehler_differential.map R S A B (kaehler_differential.D R A x) =
kaehler_differential.D S B (algebra_map A B x) | derivation.congr_fun (kaehler_differential.map_comp_der R S A B) x | lemma | kaehler_differential.map_D | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"algebra_map",
"derivation.congr_fun",
"kaehler_differential.D",
"kaehler_differential.map",
"kaehler_differential.map_comp_der"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.map_surjective_of_surjective
(h : function.surjective (algebra_map A B)) :
function.surjective (kaehler_differential.map R S A B) | begin
rw [← linear_map.range_eq_top, _root_.eq_top_iff, ← @submodule.restrict_scalars_top B A,
← kaehler_differential.span_range_derivation, submodule.restrict_scalars_span _ _ h,
submodule.span_le],
rintros _ ⟨x, rfl⟩,
obtain ⟨y, rfl⟩ := h x,
rw ← kaehler_differential.map_D R S A B,
exact ⟨_, rfl⟩,
e... | lemma | kaehler_differential.map_surjective_of_surjective | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"algebra_map",
"kaehler_differential.map",
"kaehler_differential.map_D",
"kaehler_differential.span_range_derivation",
"linear_map.range_eq_top",
"submodule.restrict_scalars_span",
"submodule.restrict_scalars_top",
"submodule.span_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kaehler_differential.map_base_change : B ⊗[A] Ω[A⁄R] →ₗ[B] Ω[B⁄R] | (tensor_product.is_base_change A Ω[A⁄R] B).lift (kaehler_differential.map R R A B) | def | kaehler_differential.map_base_change | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"kaehler_differential.map",
"lift",
"tensor_product.is_base_change"
] | The lift of the map `Ω[A⁄R] →ₗ[A] Ω[B⁄R]` to the base change along `A → B`.
This is the first map in the exact sequence `B ⊗[A] Ω[A⁄R] → Ω[B⁄R] → Ω[B⁄A] → 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kaehler_differential.map_base_change_tmul (x : B) (y : Ω[A⁄R]) :
kaehler_differential.map_base_change R A B (x ⊗ₜ y) =
x • kaehler_differential.map R R A B y | begin
conv_lhs { rw [← mul_one x, ← smul_eq_mul, ← tensor_product.smul_tmul', linear_map.map_smul] },
congr' 1,
exact is_base_change.lift_eq _ _ _
end | lemma | kaehler_differential.map_base_change_tmul | ring_theory | src/ring_theory/kaehler.lean | [
"ring_theory.derivation.to_square_zero",
"ring_theory.ideal.cotangent",
"ring_theory.is_tensor_product"
] | [
"is_base_change.lift_eq",
"kaehler_differential.map",
"kaehler_differential.map_base_change",
"linear_map.map_smul",
"mul_one",
"smul_eq_mul",
"tensor_product.smul_tmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
laurent_series (R : Type*) [has_zero R] | hahn_series ℤ R | abbreviation | laurent_series | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"hahn_series"
] | A `laurent_series` is implemented as a `hahn_series` with value group `ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_power_series (x : power_series R) : (x : laurent_series R) =
hahn_series.of_power_series ℤ R x | rfl | lemma | laurent_series.coe_power_series | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"hahn_series.of_power_series",
"laurent_series",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_coe_power_series (x : power_series R) (n : ℕ) :
hahn_series.coeff (x : laurent_series R) n = power_series.coeff R n x | by rw [coe_power_series, of_power_series_apply_coeff] | lemma | laurent_series.coeff_coe_power_series | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"power_series",
"power_series.coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_series_part (x : laurent_series R) : power_series R | power_series.mk (λ n, x.coeff (x.order + n)) | def | laurent_series.power_series_part | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"power_series",
"power_series.mk"
] | This is a power series that can be multiplied by an integer power of `X` to give our
Laurent series. If the Laurent series is nonzero, `power_series_part` has a nonzero
constant term. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_series_part_coeff (x : laurent_series R) (n : ℕ) :
power_series.coeff R n x.power_series_part = x.coeff (x.order + n) | power_series.coeff_mk _ _ | lemma | laurent_series.power_series_part_coeff | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"power_series.coeff",
"power_series.coeff_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_series_part_zero : power_series_part (0 : laurent_series R) = 0 | by { ext, simp } | lemma | laurent_series.power_series_part_zero | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_series_part_eq_zero (x : laurent_series R) :
x.power_series_part = 0 ↔ x = 0 | begin
split,
{ contrapose!,
intro h,
rw [power_series.ext_iff, not_forall],
refine ⟨0, _⟩,
simp [coeff_order_ne_zero h] },
{ rintro rfl,
simp }
end | lemma | laurent_series.power_series_part_eq_zero | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"not_forall",
"power_series.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_order_mul_power_series_part (x : laurent_series R) :
(single x.order 1 : laurent_series R) * x.power_series_part = x | begin
ext n,
rw [← sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul],
by_cases h : x.order ≤ n,
{ rw [int.eq_nat_abs_of_zero_le (sub_nonneg_of_le h), coeff_coe_power_series,
power_series_part_coeff, ← int.eq_nat_abs_of_zero_le (sub_nonneg_of_le h),
add_sub_cancel'_right] },
... | lemma | laurent_series.single_order_mul_power_series_part | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"function.embedding.coe_fn_mk",
"laurent_series",
"one_mul",
"rel_embedding.coe_fn_mk",
"set.mem_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_power_series_power_series_part (x : laurent_series R) :
of_power_series ℤ R x.power_series_part = single (-x.order) 1 * x | begin
refine eq.trans _ (congr rfl x.single_order_mul_power_series_part),
rw [← mul_assoc, single_mul_single, neg_add_self, mul_one, ← C_apply, C_one, one_mul,
coe_power_series],
end | lemma | laurent_series.of_power_series_power_series_part | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"mul_assoc",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_algebra_map [comm_semiring R] :
⇑(algebra_map (power_series R) (laurent_series R)) = hahn_series.of_power_series ℤ R | rfl | lemma | laurent_series.coe_algebra_map | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"algebra_map",
"comm_semiring",
"hahn_series.of_power_series",
"laurent_series",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_power_series_localization [comm_ring R] :
is_localization (submonoid.powers (power_series.X : power_series R)) (laurent_series R) | { map_units := (begin rintro ⟨_, n, rfl⟩,
refine ⟨⟨single (n : ℤ) 1, single (-n : ℤ) 1, _, _⟩, _⟩,
{ simp only [single_mul_single, mul_one, add_right_neg],
refl },
{ simp only [single_mul_single, mul_one, add_left_neg],
refl },
{ simp } end),
surj := (begin intro z,
by_cases h : 0 ≤ z.o... | instance | laurent_series.of_power_series_localization | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"comm_ring",
"finsupp.single_add",
"is_localization",
"laurent_series",
"linear_map.map_zero",
"mul_comm",
"mul_one",
"mv_power_series.coeff_add_monomial_mul",
"one_mul",
"power_series",
"power_series.X",
"power_series.X_pow_eq",
"power_series.coeff",
"power_series.ext_iff",
"power_serie... | The localization map from power series to Laurent series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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