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 |
|---|---|---|---|---|---|---|---|---|---|---|
coe_bot : ((⊥ : homogeneous_ideal 𝒜) : set A) = 0 | rfl | lemma | homogeneous_ideal.coe_bot | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (I J : homogeneous_ideal 𝒜) : ↑(I ⊔ J) = (I + J : set A) | submodule.coe_sup _ _ | lemma | homogeneous_ideal.coe_sup | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal",
"submodule.coe_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (I J : homogeneous_ideal 𝒜) : (↑(I ⊓ J) : set A) = I ∩ J | rfl | lemma | homogeneous_ideal.coe_inf | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_top : (⊤ : homogeneous_ideal 𝒜).to_ideal = (⊤ : ideal A) | rfl | lemma | homogeneous_ideal.to_ideal_top | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_bot : (⊥ : homogeneous_ideal 𝒜).to_ideal = (⊥ : ideal A) | rfl | lemma | homogeneous_ideal.to_ideal_bot | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_sup (I J : homogeneous_ideal 𝒜) :
(I ⊔ J).to_ideal = I.to_ideal ⊔ J.to_ideal | rfl | lemma | homogeneous_ideal.to_ideal_sup | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_inf (I J : homogeneous_ideal 𝒜) :
(I ⊓ J).to_ideal = I.to_ideal ⊓ J.to_ideal | rfl | lemma | homogeneous_ideal.to_ideal_inf | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_Sup (ℐ : set (homogeneous_ideal 𝒜)) :
(Sup ℐ).to_ideal = ⨆ s ∈ ℐ, to_ideal s | rfl | lemma | homogeneous_ideal.to_ideal_Sup | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_Inf (ℐ : set (homogeneous_ideal 𝒜)) :
(Inf ℐ).to_ideal = ⨅ s ∈ ℐ, to_ideal s | rfl | lemma | homogeneous_ideal.to_ideal_Inf | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_supr {κ : Sort*} (s : κ → homogeneous_ideal 𝒜) :
(⨆ i, s i).to_ideal = ⨆ i, (s i).to_ideal | by rw [supr, to_ideal_Sup, supr_range] | lemma | homogeneous_ideal.to_ideal_supr | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal",
"supr",
"supr_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_infi {κ : Sort*} (s : κ → homogeneous_ideal 𝒜) :
(⨅ i, s i).to_ideal = ⨅ i, (s i).to_ideal | by rw [infi, to_ideal_Inf, infi_range] | lemma | homogeneous_ideal.to_ideal_infi | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal",
"infi",
"infi_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_supr₂ {κ : Sort*} {κ' : κ → Sort*} (s : Π i, κ' i → homogeneous_ideal 𝒜) :
(⨆ i j, s i j).to_ideal = ⨆ i j, (s i j).to_ideal | by simp_rw to_ideal_supr | lemma | homogeneous_ideal.to_ideal_supr₂ | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_infi₂ {κ : Sort*} {κ' : κ → Sort*} (s : Π i, κ' i → homogeneous_ideal 𝒜) :
(⨅ i j, s i j).to_ideal = ⨅ i j, (s i j).to_ideal | by simp_rw to_ideal_infi | lemma | homogeneous_ideal.to_ideal_infi₂ | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff (I : homogeneous_ideal 𝒜) : I = ⊤ ↔ I.to_ideal = ⊤ | to_ideal_injective.eq_iff.symm | lemma | homogeneous_ideal.eq_top_iff | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"eq_top_iff",
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_bot_iff (I : homogeneous_ideal 𝒜) : I = ⊥ ↔ I.to_ideal = ⊥ | to_ideal_injective.eq_iff.symm | lemma | homogeneous_ideal.eq_bot_iff | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"eq_bot_iff",
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ideal_add (I J : homogeneous_ideal 𝒜) :
(I + J).to_ideal = I.to_ideal + J.to_ideal | rfl | lemma | homogeneous_ideal.to_ideal_add | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous.mul {I J : ideal A}
(HI : I.is_homogeneous 𝒜) (HJ : J.is_homogeneous 𝒜) : (I * J).is_homogeneous 𝒜 | begin
rw ideal.is_homogeneous.iff_exists at HI HJ ⊢,
obtain ⟨⟨s₁, rfl⟩, ⟨s₂, rfl⟩⟩ := ⟨HI, HJ⟩,
rw ideal.span_mul_span',
exact ⟨s₁ * s₂, congr_arg _ $ (set.image_mul (homogeneous_submonoid 𝒜).subtype).symm⟩,
end | lemma | ideal.is_homogeneous.mul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"ideal",
"ideal.is_homogeneous.iff_exists",
"ideal.span_mul_span'",
"set.image_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_ideal.to_ideal_mul (I J : homogeneous_ideal 𝒜) :
(I * J).to_ideal = I.to_ideal * J.to_ideal | rfl | lemma | homogeneous_ideal.to_ideal_mul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_core.gc : galois_connection to_ideal (ideal.homogeneous_core 𝒜) | λ I J, ⟨
λ H, I.to_ideal_homogeneous_core_eq_self ▸ ideal.homogeneous_core_mono 𝒜 H,
λ H, le_trans H (ideal.homogeneous_core'_le _ _)⟩ | lemma | ideal.homogeneous_core.gc | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"galois_connection",
"ideal.homogeneous_core",
"ideal.homogeneous_core'_le",
"ideal.homogeneous_core_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_core.gi : galois_coinsertion to_ideal (ideal.homogeneous_core 𝒜) | { choice := λ I HI,
⟨I, le_antisymm (I.to_ideal_homogeneous_core_le 𝒜) HI ▸ homogeneous_ideal.is_homogeneous _⟩,
gc := ideal.homogeneous_core.gc 𝒜,
u_l_le := λ I, ideal.homogeneous_core'_le _ _,
choice_eq := λ I H, le_antisymm H (I.to_ideal_homogeneous_core_le _) } | def | ideal.homogeneous_core.gi | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"galois_coinsertion",
"homogeneous_ideal.is_homogeneous",
"ideal.homogeneous_core",
"ideal.homogeneous_core'_le",
"ideal.homogeneous_core.gc"
] | `to_ideal : homogeneous_ideal 𝒜 → ideal A` and `ideal.homogeneous_core 𝒜` forms a galois
coinsertion | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.homogeneous_core_eq_Sup :
I.homogeneous_core 𝒜 = Sup {J : homogeneous_ideal 𝒜 | J.to_ideal ≤ I} | eq.symm $ is_lub.Sup_eq $ (ideal.homogeneous_core.gc 𝒜).is_greatest_u.is_lub | lemma | ideal.homogeneous_core_eq_Sup | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal",
"ideal.homogeneous_core.gc",
"is_lub.Sup_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_core'_eq_Sup :
I.homogeneous_core' 𝒜 = Sup {J : ideal A | J.is_homogeneous 𝒜 ∧ J ≤ I} | begin
refine (is_lub.Sup_eq _).symm,
apply is_greatest.is_lub,
have coe_mono : monotone (to_ideal : homogeneous_ideal 𝒜 → ideal A) := λ x y, id,
convert coe_mono.map_is_greatest (ideal.homogeneous_core.gc 𝒜).is_greatest_u using 1,
ext,
rw [mem_image, mem_set_of_eq],
refine ⟨λ hI, ⟨⟨x, hI.1⟩, ⟨hI.2, rfl⟩... | lemma | ideal.homogeneous_core'_eq_Sup | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal",
"ideal",
"ideal.homogeneous_core.gc",
"is_greatest.is_lub",
"is_lub.Sup_eq",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_hull : homogeneous_ideal 𝒜 | ⟨ideal.span {r : A | ∃ (i : ι) (x : I), (direct_sum.decompose 𝒜 (x : A) i : A) = r}, begin
refine ideal.is_homogeneous_span _ _ (λ x hx, _),
obtain ⟨i, x, rfl⟩ := hx,
apply set_like.is_homogeneous_coe
end⟩ | def | ideal.homogeneous_hull | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"direct_sum.decompose",
"homogeneous_ideal",
"ideal.is_homogeneous_span",
"set_like.is_homogeneous_coe"
] | For any `I : ideal A`, not necessarily homogeneous, `I.homogeneous_hull 𝒜` is
the smallest homogeneous ideal containing `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.le_to_ideal_homogeneous_hull :
I ≤ (ideal.homogeneous_hull 𝒜 I).to_ideal | begin
intros r hr,
classical,
rw [←direct_sum.sum_support_decompose 𝒜 r],
refine ideal.sum_mem _ _, intros j hj,
apply ideal.subset_span, use j, use ⟨r, hr⟩, refl,
end | lemma | ideal.le_to_ideal_homogeneous_hull | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"ideal.homogeneous_hull",
"ideal.subset_span",
"ideal.sum_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_hull_mono : monotone (ideal.homogeneous_hull 𝒜) | λ I J I_le_J,
begin
apply ideal.span_mono,
rintros r ⟨hr1, ⟨x, hx⟩, rfl⟩,
refine ⟨hr1, ⟨⟨x, I_le_J hx⟩, rfl⟩⟩,
end | lemma | ideal.homogeneous_hull_mono | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"ideal.homogeneous_hull",
"ideal.span_mono",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous.to_ideal_homogeneous_hull_eq_self (h : I.is_homogeneous 𝒜) :
(ideal.homogeneous_hull 𝒜 I).to_ideal = I | begin
apply le_antisymm _ (ideal.le_to_ideal_homogeneous_hull _ _),
apply (ideal.span_le).2,
rintros _ ⟨i, x, rfl⟩,
exact h _ x.prop,
end | lemma | ideal.is_homogeneous.to_ideal_homogeneous_hull_eq_self | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"ideal.homogeneous_hull",
"ideal.le_to_ideal_homogeneous_hull",
"ideal.span_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_ideal.homogeneous_hull_to_ideal_eq_self (I : homogeneous_ideal 𝒜) :
I.to_ideal.homogeneous_hull 𝒜 = I | homogeneous_ideal.to_ideal_injective $ I.is_homogeneous.to_ideal_homogeneous_hull_eq_self | lemma | homogeneous_ideal.homogeneous_hull_to_ideal_eq_self | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal",
"homogeneous_ideal.to_ideal_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.to_ideal_homogeneous_hull_eq_supr :
(I.homogeneous_hull 𝒜).to_ideal = ⨆ i, ideal.span (graded_ring.proj 𝒜 i '' I) | begin
rw ←ideal.span_Union,
apply congr_arg ideal.span _,
ext1,
simp only [set.mem_Union, set.mem_image, mem_set_of_eq, graded_ring.proj_apply,
set_like.exists, exists_prop, subtype.coe_mk, set_like.mem_coe],
end | lemma | ideal.to_ideal_homogeneous_hull_eq_supr | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"exists_prop",
"graded_ring.proj",
"graded_ring.proj_apply",
"ideal.span",
"set.mem_Union",
"set.mem_image",
"set_like.mem_coe",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_hull_eq_supr :
(I.homogeneous_hull 𝒜) =
⨆ i, ⟨ideal.span (graded_ring.proj 𝒜 i '' I), ideal.is_homogeneous_span 𝒜 _
(by {rintros _ ⟨x, -, rfl⟩, apply set_like.is_homogeneous_coe})⟩ | by { ext1, rw [ideal.to_ideal_homogeneous_hull_eq_supr, to_ideal_supr], refl } | lemma | ideal.homogeneous_hull_eq_supr | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"graded_ring.proj",
"ideal.is_homogeneous_span",
"ideal.to_ideal_homogeneous_hull_eq_supr",
"set_like.is_homogeneous_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_hull.gc : galois_connection (ideal.homogeneous_hull 𝒜) to_ideal | λ I J, ⟨
le_trans (ideal.le_to_ideal_homogeneous_hull _ _),
λ H, J.homogeneous_hull_to_ideal_eq_self ▸ ideal.homogeneous_hull_mono 𝒜 H⟩ | lemma | ideal.homogeneous_hull.gc | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"galois_connection",
"ideal.homogeneous_hull",
"ideal.homogeneous_hull_mono",
"ideal.le_to_ideal_homogeneous_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_hull.gi : galois_insertion (ideal.homogeneous_hull 𝒜) to_ideal | { choice := λ I H, ⟨I, le_antisymm H (I.le_to_ideal_homogeneous_hull 𝒜) ▸ is_homogeneous _⟩,
gc := ideal.homogeneous_hull.gc 𝒜,
le_l_u := λ I, ideal.le_to_ideal_homogeneous_hull _ _,
choice_eq := λ I H, le_antisymm (I.le_to_ideal_homogeneous_hull 𝒜) H} | def | ideal.homogeneous_hull.gi | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"galois_insertion",
"ideal.homogeneous_hull",
"ideal.homogeneous_hull.gc",
"ideal.le_to_ideal_homogeneous_hull"
] | `ideal.homogeneous_hull 𝒜` and `to_ideal : homogeneous_ideal 𝒜 → ideal A` form a galois
insertion | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.homogeneous_hull_eq_Inf (I : ideal A) :
ideal.homogeneous_hull 𝒜 I = Inf { J : homogeneous_ideal 𝒜 | I ≤ J.to_ideal } | eq.symm $ is_glb.Inf_eq $ (ideal.homogeneous_hull.gc 𝒜).is_least_l.is_glb | lemma | ideal.homogeneous_hull_eq_Inf | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal",
"ideal",
"ideal.homogeneous_hull",
"ideal.homogeneous_hull.gc",
"is_glb.Inf_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_ideal.irrelevant : homogeneous_ideal 𝒜 | ⟨(graded_ring.proj_zero_ring_hom 𝒜).ker, λ i r (hr : (decompose 𝒜 r 0 : A) = 0), begin
change (decompose 𝒜 (decompose 𝒜 r _ : A) 0 : A) = 0,
by_cases h : i = 0,
{ rw [h, hr, decompose_zero, zero_apply, zero_mem_class.coe_zero] },
{ rw [decompose_of_mem_ne 𝒜 (set_like.coe_mem _) h] }
end⟩ | def | homogeneous_ideal.irrelevant | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"graded_ring.proj_zero_ring_hom",
"homogeneous_ideal",
"set_like.coe_mem"
] | For a graded ring `⨁ᵢ 𝒜ᵢ` graded by a `canonically_ordered_add_monoid ι`, the irrelevant ideal
refers to `⨁_{i>0} 𝒜ᵢ`, or equivalently `{a | a₀ = 0}`. This definition is used in `Proj`
construction where `ι` is always `ℕ` so the irrelevant ideal is simply elements with `0` as
0-th coordinate.
# Future work
Here in t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_ideal.mem_irrelevant_iff (a : A) :
a ∈ homogeneous_ideal.irrelevant 𝒜 ↔ proj 𝒜 0 a = 0 | iff.rfl | lemma | homogeneous_ideal.mem_irrelevant_iff | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_ideal.irrelevant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_ideal.to_ideal_irrelevant :
(homogeneous_ideal.irrelevant 𝒜).to_ideal = (graded_ring.proj_zero_ring_hom 𝒜).ker | rfl | lemma | homogeneous_ideal.to_ideal_irrelevant | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_ideal.lean | [
"ring_theory.ideal.basic",
"ring_theory.ideal.operations",
"linear_algebra.finsupp",
"ring_theory.graded_algebra.basic"
] | [
"graded_ring.proj_zero_ring_hom",
"homogeneous_ideal.irrelevant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_denom_same_deg | (deg : ι)
(num denom : 𝒜 deg)
(denom_mem : (denom : A) ∈ x) | structure | homogeneous_localization.num_denom_same_deg | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"num"
] | Let `x` be a submonoid of `A`, then `num_denom_same_deg 𝒜 x` is a structure with a numerator and a
denominator with same grading such that the denominator is contained in `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {c1 c2 : num_denom_same_deg 𝒜 x} (hdeg : c1.deg = c2.deg)
(hnum : (c1.num : A) = c2.num) (hdenom : (c1.denom : A) = c2.denom) :
c1 = c2 | begin
rcases c1 with ⟨i1, ⟨n1, hn1⟩, ⟨d1, hd1⟩, h1⟩,
rcases c2 with ⟨i2, ⟨n2, hn2⟩, ⟨d2, hd2⟩, h2⟩,
dsimp only [subtype.coe_mk] at *,
simp only, exact ⟨hdeg, by subst hdeg; subst hnum, by subst hdeg; subst hdenom⟩,
end | lemma | homogeneous_localization.num_denom_same_deg.ext | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deg_one : (1 : num_denom_same_deg 𝒜 x).deg = 0 | rfl | lemma | homogeneous_localization.num_denom_same_deg.deg_one | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_one : ((1 : num_denom_same_deg 𝒜 x).num : A) = 1 | rfl | lemma | homogeneous_localization.num_denom_same_deg.num_one | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_one : ((1 : num_denom_same_deg 𝒜 x).denom : A) = 1 | rfl | lemma | homogeneous_localization.num_denom_same_deg.denom_one | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deg_zero : (0 : num_denom_same_deg 𝒜 x).deg = 0 | rfl | lemma | homogeneous_localization.num_denom_same_deg.deg_zero | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_zero : (0 : num_denom_same_deg 𝒜 x).num = 0 | rfl | lemma | homogeneous_localization.num_denom_same_deg.num_zero | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_zero : ((0 : num_denom_same_deg 𝒜 x).denom : A) = 1 | rfl | lemma | homogeneous_localization.num_denom_same_deg.denom_zero | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deg_mul (c1 c2 : num_denom_same_deg 𝒜 x) : (c1 * c2).deg = c1.deg + c2.deg | rfl | lemma | homogeneous_localization.num_denom_same_deg.deg_mul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_mul (c1 c2 : num_denom_same_deg 𝒜 x) :
((c1 * c2).num : A) = c1.num * c2.num | rfl | lemma | homogeneous_localization.num_denom_same_deg.num_mul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_mul (c1 c2 : num_denom_same_deg 𝒜 x) :
((c1 * c2).denom : A) = c1.denom * c2.denom | rfl | lemma | homogeneous_localization.num_denom_same_deg.denom_mul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deg_add (c1 c2 : num_denom_same_deg 𝒜 x) : (c1 + c2).deg = c1.deg + c2.deg | rfl | lemma | homogeneous_localization.num_denom_same_deg.deg_add | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_add (c1 c2 : num_denom_same_deg 𝒜 x) :
((c1 + c2).num : A) = c1.denom * c2.num + c2.denom * c1.num | rfl | lemma | homogeneous_localization.num_denom_same_deg.num_add | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_add (c1 c2 : num_denom_same_deg 𝒜 x) :
((c1 + c2).denom : A) = c1.denom * c2.denom | rfl | lemma | homogeneous_localization.num_denom_same_deg.denom_add | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deg_neg (c : num_denom_same_deg 𝒜 x) : (-c).deg = c.deg | rfl | lemma | homogeneous_localization.num_denom_same_deg.deg_neg | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_neg (c : num_denom_same_deg 𝒜 x) : ((-c).num : A) = -c.num | rfl | lemma | homogeneous_localization.num_denom_same_deg.num_neg | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_neg (c : num_denom_same_deg 𝒜 x) : ((-c).denom : A) = c.denom | rfl | lemma | homogeneous_localization.num_denom_same_deg.denom_neg | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deg_pow (c : num_denom_same_deg 𝒜 x) (n : ℕ) : (c ^ n).deg = n • c.deg | rfl | lemma | homogeneous_localization.num_denom_same_deg.deg_pow | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_pow (c : num_denom_same_deg 𝒜 x) (n : ℕ) : ((c ^ n).num : A) = c.num ^ n | rfl | lemma | homogeneous_localization.num_denom_same_deg.num_pow | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_pow (c : num_denom_same_deg 𝒜 x) (n : ℕ) :
((c ^ n).denom : A) = c.denom ^ n | rfl | lemma | homogeneous_localization.num_denom_same_deg.denom_pow | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deg_smul (c : num_denom_same_deg 𝒜 x) (m : α) : (m • c).deg = c.deg | rfl | lemma | homogeneous_localization.num_denom_same_deg.deg_smul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_smul (c : num_denom_same_deg 𝒜 x) (m : α) : ((m • c).num : A) = m • c.num | rfl | lemma | homogeneous_localization.num_denom_same_deg.num_smul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"num"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_smul (c : num_denom_same_deg 𝒜 x) (m : α) :
((m • c).denom : A) = c.denom | rfl | lemma | homogeneous_localization.num_denom_same_deg.denom_smul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding (p : num_denom_same_deg 𝒜 x) : at x | localization.mk p.num ⟨p.denom, p.denom_mem⟩ | def | homogeneous_localization.num_denom_same_deg.embedding | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"embedding",
"localization.mk"
] | For `x : prime ideal of A` and any `p : num_denom_same_deg 𝒜 x`, or equivalent a numerator and a
denominator of the same degree, we get an element `p.num / p.denom` of `Aₓ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_localization : Type* | quotient (setoid.ker $ homogeneous_localization.num_denom_same_deg.embedding 𝒜 x) | def | homogeneous_localization | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization.num_denom_same_deg.embedding",
"setoid.ker"
] | For `x : prime ideal of A`, `homogeneous_localization 𝒜 x` is `num_denom_same_deg 𝒜 x` modulo the
kernel of `embedding 𝒜 x`. This is essentially the subring of `Aₓ` where the numerator and
denominator share the same grading. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
val (y : homogeneous_localization 𝒜 x) : at x | quotient.lift_on' y (num_denom_same_deg.embedding 𝒜 x) $ λ _ _, id | def | homogeneous_localization.val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"quotient.lift_on'"
] | View an element of `homogeneous_localization 𝒜 x` as an element of `Aₓ` by forgetting that the
numerator and denominator are of the same grading. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
val_mk' (i : num_denom_same_deg 𝒜 x) :
val (quotient.mk' i) = localization.mk i.num ⟨i.denom, i.denom_mem⟩ | rfl | lemma | homogeneous_localization.val_mk' | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"localization.mk",
"quotient.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val_injective :
function.injective (@homogeneous_localization.val _ _ _ _ _ _ _ _ 𝒜 _ x) | λ a b, quotient.rec_on_subsingleton₂' a b $ λ a b h, quotient.sound' h | lemma | homogeneous_localization.val_injective | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization.val",
"quotient.rec_on_subsingleton₂'",
"quotient.sound'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pow : has_pow (homogeneous_localization 𝒜 x) ℕ | { pow := λ z n, (quotient.map' (^ n)
(λ c1 c2 (h : localization.mk _ _ = localization.mk _ _), begin
change localization.mk _ _ = localization.mk _ _,
simp only [num_pow, denom_pow],
convert congr_arg (λ z, z ^ n) h;
erw localization.mk_pow;
refl,
end) : homogeneous_localization 𝒜... | instance | homogeneous_localization.has_pow | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"localization.mk",
"localization.mk_pow",
"quotient.map'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_val (y : homogeneous_localization 𝒜 x) (n : α) :
(n • y).val = n • y.val | begin
induction y using quotient.induction_on,
unfold homogeneous_localization.val has_smul.smul,
simp only [quotient.lift_on₂'_mk, quotient.lift_on'_mk],
change localization.mk _ _ = n • localization.mk _ _,
dsimp only,
rw localization.smul_mk,
congr' 1,
end | lemma | homogeneous_localization.smul_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"homogeneous_localization.val",
"localization.mk",
"localization.smul_mk",
"quotient.lift_on'_mk",
"quotient.lift_on₂'_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq :
(0 : homogeneous_localization 𝒜 x) = quotient.mk' 0 | rfl | lemma | homogeneous_localization.zero_eq | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"quotient.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq :
(1 : homogeneous_localization 𝒜 x) = quotient.mk' 1 | rfl | lemma | homogeneous_localization.one_eq | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"quotient.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_val : (0 : homogeneous_localization 𝒜 x).val = 0 | localization.mk_zero _ | lemma | homogeneous_localization.zero_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"localization.mk_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_val : (1 : homogeneous_localization 𝒜 x).val = 1 | localization.mk_one | lemma | homogeneous_localization.one_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"localization.mk_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val (y1 y2 : homogeneous_localization 𝒜 x) :
(y1 + y2).val = y1.val + y2.val | begin
induction y1 using quotient.induction_on,
induction y2 using quotient.induction_on,
unfold homogeneous_localization.val has_add.add,
simp only [quotient.lift_on₂'_mk, quotient.lift_on'_mk],
change localization.mk _ _ = localization.mk _ _ + localization.mk _ _,
dsimp only,
rw [localization.add_mk],
... | lemma | homogeneous_localization.add_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"homogeneous_localization.val",
"localization.add_mk",
"localization.mk",
"quotient.lift_on'_mk",
"quotient.lift_on₂'_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_val (y1 y2 : homogeneous_localization 𝒜 x) :
(y1 * y2).val = y1.val * y2.val | begin
induction y1 using quotient.induction_on,
induction y2 using quotient.induction_on,
unfold homogeneous_localization.val has_mul.mul,
simp only [quotient.lift_on₂'_mk, quotient.lift_on'_mk],
change localization.mk _ _ = localization.mk _ _ * localization.mk _ _,
dsimp only,
rw [localization.mk_mul],
... | lemma | homogeneous_localization.mul_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"homogeneous_localization.val",
"localization.mk",
"localization.mk_mul",
"quotient.lift_on'_mk",
"quotient.lift_on₂'_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_val (y : homogeneous_localization 𝒜 x) :
(-y).val = -y.val | begin
induction y using quotient.induction_on,
unfold homogeneous_localization.val has_neg.neg,
simp only [quotient.lift_on₂'_mk, quotient.lift_on'_mk],
change localization.mk _ _ = - localization.mk _ _,
dsimp only,
rw [localization.neg_mk],
refl,
end | lemma | homogeneous_localization.neg_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"homogeneous_localization.val",
"localization.mk",
"localization.neg_mk",
"quotient.lift_on'_mk",
"quotient.lift_on₂'_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_val (y1 y2 : homogeneous_localization 𝒜 x) :
(y1 - y2).val = y1.val - y2.val | by rw [show y1 - y2 = y1 + (-y2), from rfl, add_val, neg_val]; refl | lemma | homogeneous_localization.sub_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_val (y : homogeneous_localization 𝒜 x) (n : ℕ) :
(y ^ n).val = y.val ^ n | begin
induction y using quotient.induction_on,
unfold homogeneous_localization.val has_pow.pow,
simp only [quotient.lift_on₂'_mk, quotient.lift_on'_mk],
change localization.mk _ _ = (localization.mk _ _) ^ n,
rw localization.mk_pow,
dsimp only,
congr' 1,
end | lemma | homogeneous_localization.pow_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"homogeneous_localization.val",
"localization.mk",
"localization.mk_pow",
"quotient.lift_on'_mk",
"quotient.lift_on₂'_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_val (n : ℕ) : (n : homogeneous_localization 𝒜 x).val = n | show val (nat.unary_cast n) = _, by induction n; simp [nat.unary_cast, zero_val, one_val, *] | lemma | homogeneous_localization.nat_cast_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"nat.unary_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_val (n : ℤ) : (n : homogeneous_localization 𝒜 x).val = n | show val (int.cast_def n) = _, by cases n; simp [int.cast_def, zero_val, one_val, *] | lemma | homogeneous_localization.int_cast_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"int.cast_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogenous_localization_comm_ring : comm_ring (homogeneous_localization 𝒜 x) | (homogeneous_localization.val_injective x).comm_ring _ zero_val one_val add_val mul_val neg_val
sub_val (λ z n, smul_val x z n) (λ z n, smul_val x z n) pow_val nat_cast_val int_cast_val | instance | homogeneous_localization.homogenous_localization_comm_ring | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"comm_ring",
"homogeneous_localization",
"homogeneous_localization.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_localization_algebra :
algebra (homogeneous_localization 𝒜 x) (localization x) | { smul := λ p q, p.val * q,
to_fun := val,
map_one' := one_val,
map_mul' := mul_val,
map_zero' := zero_val,
map_add' := add_val,
commutes' := λ p q, mul_comm _ _,
smul_def' := λ p q, rfl } | instance | homogeneous_localization.homogeneous_localization_algebra | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"algebra",
"homogeneous_localization",
"localization",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num (f : homogeneous_localization 𝒜 x) : A | (quotient.out' f).num | def | homogeneous_localization.num | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"num",
"quotient.out'"
] | numerator of an element in `homogeneous_localization x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
denom (f : homogeneous_localization 𝒜 x) : A | (quotient.out' f).denom | def | homogeneous_localization.denom | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"quotient.out'"
] | denominator of an element in `homogeneous_localization x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deg (f : homogeneous_localization 𝒜 x) : ι | (quotient.out' f).deg | def | homogeneous_localization.deg | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"quotient.out'"
] | For an element in `homogeneous_localization x`, degree is the natural number `i` such that
`𝒜 i` contains both numerator and denominator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
denom_mem (f : homogeneous_localization 𝒜 x) :
f.denom ∈ x | (quotient.out' f).denom_mem | lemma | homogeneous_localization.denom_mem | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"quotient.out'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
num_mem_deg (f : homogeneous_localization 𝒜 x) : f.num ∈ 𝒜 f.deg | (quotient.out' f).num.2 | lemma | homogeneous_localization.num_mem_deg | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"quotient.out'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denom_mem_deg (f : homogeneous_localization 𝒜 x) : f.denom ∈ 𝒜 f.deg | (quotient.out' f).denom.2 | lemma | homogeneous_localization.denom_mem_deg | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"quotient.out'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_num_div_denom (f : homogeneous_localization 𝒜 x) :
f.val = localization.mk f.num ⟨f.denom, f.denom_mem⟩ | begin
have := (quotient.out_eq' f),
apply_fun homogeneous_localization.val at this,
rw ← this,
unfold homogeneous_localization.val,
simp only [quotient.lift_on'_mk'],
refl,
end | lemma | homogeneous_localization.eq_num_div_denom | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"homogeneous_localization.val",
"localization.mk",
"quotient.lift_on'_mk'",
"quotient.out_eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff_val (f g : homogeneous_localization 𝒜 x) : f = g ↔ f.val = g.val | { mp := λ h, h ▸ rfl,
mpr := λ h, begin
induction f using quotient.induction_on,
induction g using quotient.induction_on,
rw quotient.eq,
unfold homogeneous_localization.val at h,
simpa only [quotient.lift_on'_mk] using h,
end } | lemma | homogeneous_localization.ext_iff_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"homogeneous_localization.val",
"quotient.eq",
"quotient.lift_on'_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
at_prime | homogeneous_localization 𝒜 𝔭.prime_compl | abbreviation | homogeneous_localization.at_prime | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization"
] | Localizing a ring homogeneously at a prime ideal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_iff_is_unit_val (f : homogeneous_localization.at_prime 𝒜 𝔭) :
is_unit f.val ↔ is_unit f | ⟨λ h1, begin
rcases h1 with ⟨⟨a, b, eq0, eq1⟩, (eq2 : a = f.val)⟩,
rw eq2 at eq0 eq1,
clear' a eq2,
induction b using localization.induction_on with data,
rcases data with ⟨a, ⟨b, hb⟩⟩,
dsimp only at eq0 eq1,
have b_f_denom_not_mem : b * f.denom ∈ 𝔭.prime_compl := λ r, or.elim
(ideal.is_prime.mem_or_... | lemma | homogeneous_localization.is_unit_iff_is_unit_val | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization.at_prime",
"ideal.is_prime.mem_or_mem",
"ideal.mul_mem_left",
"is_localization.eq",
"is_unit",
"localization.at_prime",
"localization.induction_on",
"localization.mk",
"localization.mk_eq_mk'",
"localization.mk_mul",
"localization.mk_self",
"mul_comm",
"mul_one",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
away | homogeneous_localization 𝒜 (submonoid.powers f) | abbreviation | homogeneous_localization.away | ring_theory.graded_algebra | src/ring_theory/graded_algebra/homogeneous_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.graded_algebra.basic"
] | [
"homogeneous_localization",
"submonoid.powers"
] | Localising away from powers of `f` homogeneously. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.is_homogeneous.is_prime_of_homogeneous_mem_or_mem
{I : ideal A} (hI : I.is_homogeneous 𝒜) (I_ne_top : I ≠ ⊤)
(homogeneous_mem_or_mem : ∀ {x y : A},
is_homogeneous 𝒜 x → is_homogeneous 𝒜 y → (x * y ∈ I → x ∈ I ∨ y ∈ I)) :
ideal.is_prime I | ⟨I_ne_top, begin
intros x y hxy,
by_contradiction rid,
obtain ⟨rid₁, rid₂⟩ := not_or_distrib.mp rid,
/-
The idea of the proof is the following :
since `x * y ∈ I` and `I` homogeneous, then `proj i (x * y) ∈ I` for any `i : ι`.
Then consider two sets `{i ∈ x.support | xᵢ ∉ I}` and `{j ∈ y.support | yⱼ ∉ J}... | lemma | ideal.is_homogeneous.is_prime_of_homogeneous_mem_or_mem | ring_theory.graded_algebra | src/ring_theory/graded_algebra/radical.lean | [
"ring_theory.graded_algebra.homogeneous_ideal"
] | [
"by_contradiction",
"direct_sum.coe_mul_apply",
"direct_sum.decompose_mul",
"filter",
"ideal",
"ideal.is_prime",
"ideal.mul_mem_left",
"ideal.mul_mem_right",
"ideal.sub_mem",
"ideal.sum_mem",
"not_and",
"not_not",
"prod.mk.inj_iff",
"set_like.coe_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous.is_prime_iff {I : ideal A} (h : I.is_homogeneous 𝒜) :
I.is_prime ↔
(I ≠ ⊤) ∧
∀ {x y : A}, set_like.is_homogeneous 𝒜 x → set_like.is_homogeneous 𝒜 y
→ (x * y ∈ I → x ∈ I ∨ y ∈ I) | ⟨λ HI,
⟨ne_of_apply_ne _ HI.ne_top, λ x y hx hy hxy, ideal.is_prime.mem_or_mem HI hxy⟩,
λ ⟨I_ne_top, homogeneous_mem_or_mem⟩,
h.is_prime_of_homogeneous_mem_or_mem I_ne_top @homogeneous_mem_or_mem⟩ | lemma | ideal.is_homogeneous.is_prime_iff | ring_theory.graded_algebra | src/ring_theory/graded_algebra/radical.lean | [
"ring_theory.graded_algebra.homogeneous_ideal"
] | [
"ideal",
"ideal.is_prime.mem_or_mem",
"set_like.is_homogeneous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_prime.homogeneous_core {I : ideal A} (h : I.is_prime) :
(I.homogeneous_core 𝒜).to_ideal.is_prime | begin
apply (ideal.homogeneous_core 𝒜 I).is_homogeneous.is_prime_of_homogeneous_mem_or_mem,
{ exact ne_top_of_le_ne_top h.ne_top (ideal.to_ideal_homogeneous_core_le 𝒜 I) },
rintros x y hx hy hxy,
have H := h.mem_or_mem (ideal.to_ideal_homogeneous_core_le 𝒜 I hxy),
refine H.imp _ _,
{ exact ideal.mem_homo... | lemma | ideal.is_prime.homogeneous_core | ring_theory.graded_algebra | src/ring_theory/graded_algebra/radical.lean | [
"ring_theory.graded_algebra.homogeneous_ideal"
] | [
"ideal",
"ideal.homogeneous_core",
"ideal.mem_homogeneous_core_of_is_homogeneous_of_mem",
"ideal.to_ideal_homogeneous_core_le",
"ne_top_of_le_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous.radical_eq {I : ideal A} (hI : I.is_homogeneous 𝒜) :
I.radical = Inf { J | J.is_homogeneous 𝒜 ∧ I ≤ J ∧ J.is_prime } | begin
rw ideal.radical_eq_Inf,
apply le_antisymm,
{ exact Inf_le_Inf (λ J, and.right), },
{ refine Inf_le_Inf_of_forall_exists_le _,
rintros J ⟨HJ₁, HJ₂⟩,
refine ⟨(J.homogeneous_core 𝒜).to_ideal, _, J.to_ideal_homogeneous_core_le _⟩,
refine ⟨homogeneous_ideal.is_homogeneous _, _, HJ₂.homogeneous_co... | lemma | ideal.is_homogeneous.radical_eq | ring_theory.graded_algebra | src/ring_theory/graded_algebra/radical.lean | [
"ring_theory.graded_algebra.homogeneous_ideal"
] | [
"Inf_le_Inf",
"Inf_le_Inf_of_forall_exists_le",
"ideal",
"ideal.homogeneous_core_mono",
"ideal.radical_eq_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous.radical {I : ideal A} (h : I.is_homogeneous 𝒜) :
I.radical.is_homogeneous 𝒜 | by { rw h.radical_eq, exact ideal.is_homogeneous.Inf (λ _, and.left) } | lemma | ideal.is_homogeneous.radical | ring_theory.graded_algebra | src/ring_theory/graded_algebra/radical.lean | [
"ring_theory.graded_algebra.homogeneous_ideal"
] | [
"ideal",
"ideal.is_homogeneous.Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_ideal.radical (I : homogeneous_ideal 𝒜) : homogeneous_ideal 𝒜 | ⟨I.to_ideal.radical, I.is_homogeneous.radical⟩ | def | homogeneous_ideal.radical | ring_theory.graded_algebra | src/ring_theory/graded_algebra/radical.lean | [
"ring_theory.graded_algebra.homogeneous_ideal"
] | [
"homogeneous_ideal"
] | The radical of a homogenous ideal, as another homogenous ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_ideal.coe_radical (I : homogeneous_ideal 𝒜) :
I.radical.to_ideal = I.to_ideal.radical | rfl | lemma | homogeneous_ideal.coe_radical | ring_theory.graded_algebra | src/ring_theory/graded_algebra/radical.lean | [
"ring_theory.graded_algebra.homogeneous_ideal"
] | [
"homogeneous_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_associated_prime : Prop | I.is_prime ∧ ∃ x : M, I = (R ∙ x).annihilator | def | is_associated_prime | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [] | `is_associated_prime I M` if the prime ideal `I` is the annihilator of some `x : M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_primes : set (ideal R) | { I | is_associated_prime I M } | def | associated_primes | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"ideal",
"is_associated_prime"
] | The set of associated primes of a module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associate_primes.mem_iff : I ∈ associated_primes R M ↔ is_associated_prime I M | iff.rfl | lemma | associate_primes.mem_iff | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"associated_primes",
"is_associated_prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_associated_prime.is_prime : I.is_prime | h.1 | lemma | is_associated_prime.is_prime | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.