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
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|---|---|---|---|---|---|---|---|---|---|---|
fg_of_fg_map {R M P : Type*} [ring R] [add_comm_group M] [module R M]
[add_comm_group P] [module R P] (f : M →ₗ[R] P) (hf : f.ker = ⊥) {N : submodule R M}
(hfn : (N.map f).fg) : N.fg | fg_of_fg_map_injective f (linear_map.ker_eq_bot.1 hf) hfn | lemma | submodule.fg_of_fg_map | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_group",
"module",
"ring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_top (N : submodule R M) : (⊤ : submodule R N).fg ↔ N.fg | ⟨λ h, N.range_subtype ▸ map_top N.subtype ▸ h.map _,
λ h, fg_of_fg_map_injective N.subtype subtype.val_injective $ by rwa [map_top, range_subtype]⟩ | lemma | submodule.fg_top | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"submodule",
"subtype.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_of_linear_equiv (e : M ≃ₗ[R] P) (h : (⊤ : submodule R P).fg) :
(⊤ : submodule R M).fg | e.symm.range ▸ map_top (e.symm : P →ₗ[R] M) ▸ h.map _ | lemma | submodule.fg_of_linear_equiv | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg.prod {sb : submodule R M} {sc : submodule R P}
(hsb : sb.fg) (hsc : sc.fg) : (sb.prod sc).fg | let ⟨tb, htb⟩ := fg_def.1 hsb, ⟨tc, htc⟩ := fg_def.1 hsc in
fg_def.2 ⟨linear_map.inl R M P '' tb ∪ linear_map.inr R M P '' tc,
(htb.1.image _).union (htc.1.image _),
by rw [linear_map.span_inl_union_inr, htb.2, htc.2]⟩ | theorem | submodule.fg.prod | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"linear_map.inr",
"linear_map.span_inl_union_inr",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_pi {ι : Type*} {M : ι → Type*} [finite ι] [Π i, add_comm_monoid (M i)]
[Π i, module R (M i)] {p : Π i, submodule R (M i)} (hsb : ∀ i, (p i).fg) :
(submodule.pi set.univ p).fg | begin
classical,
simp_rw fg_def at hsb ⊢,
choose t htf hts using hsb,
refine ⟨
⋃ i, (linear_map.single i : _ →ₗ[R] _) '' t i, set.finite_Union $ λ i, (htf i).image _, _⟩,
simp_rw [span_Union, span_image, hts, submodule.supr_map_single],
end | theorem | submodule.fg_pi | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"finite",
"linear_map.single",
"module",
"set.finite_Union",
"submodule",
"submodule.pi",
"submodule.supr_map_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_of_fg_map_of_fg_inf_ker {R M P : Type*} [ring R] [add_comm_group M] [module R M]
[add_comm_group P] [module R P] (f : M →ₗ[R] P)
{s : submodule R M} (hs1 : (s.map f).fg) (hs2 : (s ⊓ f.ker).fg) : s.fg | begin
haveI := classical.dec_eq R, haveI := classical.dec_eq M, haveI := classical.dec_eq P,
cases hs1 with t1 ht1, cases hs2 with t2 ht2,
have : ∀ y ∈ t1, ∃ x ∈ s, f x = y,
{ intros y hy,
have : y ∈ map f s, { rw ← ht1, exact subset_span hy },
rcases mem_map.1 this with ⟨x, hx1, hx2⟩,
exact ⟨x, hx1... | theorem | submodule.fg_of_fg_map_of_fg_inf_ker | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_group",
"add_smul",
"classical.dec_eq",
"finset.coe_image",
"finset.coe_union",
"finsupp.lmap_domain",
"finsupp.lmap_domain_apply",
"finsupp.lmap_domain_supported",
"finsupp.mem_span_image_iff_total",
"finsupp.total",
"finsupp.total_apply",
"linear_map.mem_ker",
"mem_map",
"modul... | If 0 → M' → M → M'' → 0 is exact and M' and M'' are
finitely generated then so is M. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fg_induction (R M : Type*) [semiring R] [add_comm_monoid M] [module R M]
(P : submodule R M → Prop)
(h₁ : ∀ x, P (submodule.span R {x})) (h₂ : ∀ M₁ M₂, P M₁ → P M₂ → P (M₁ ⊔ M₂))
(N : submodule R M) (hN : N.fg) : P N | begin
classical,
obtain ⟨s, rfl⟩ := hN,
induction s using finset.induction,
{ rw [finset.coe_empty, submodule.span_empty, ← submodule.span_zero_singleton], apply h₁ },
{ rw [finset.coe_insert, submodule.span_insert], apply h₂; apply_assumption }
end | lemma | submodule.fg_induction | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"finset.coe_empty",
"finset.coe_insert",
"finset.induction",
"module",
"semiring",
"submodule",
"submodule.span",
"submodule.span_empty",
"submodule.span_insert",
"submodule.span_zero_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_ker_comp {R M N P : Type*} [ring R] [add_comm_group M] [module R M]
[add_comm_group N] [module R N] [add_comm_group P] [module R P] (f : M →ₗ[R] N)
(g : N →ₗ[R] P) (hf1 : f.ker.fg) (hf2 : g.ker.fg) (hsur : function.surjective f) :
(g.comp f).ker.fg | begin
rw linear_map.ker_comp,
apply fg_of_fg_map_of_fg_inf_ker f,
{ rwa [submodule.map_comap_eq, linear_map.range_eq_top.2 hsur, top_inf_eq] },
{ rwa [inf_of_le_right (show f.ker ≤ (comap f g.ker), from comap_mono bot_le)] }
end | lemma | submodule.fg_ker_comp | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_group",
"bot_le",
"linear_map.ker_comp",
"module",
"ring",
"submodule.map_comap_eq",
"top_inf_eq"
] | The kernel of the composition of two linear maps is finitely generated if both kernels are and
the first morphism is surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fg_restrict_scalars {R S M : Type*} [comm_semiring R] [semiring S] [algebra R S]
[add_comm_group M] [module S M] [module R M] [is_scalar_tower R S M] (N : submodule S M)
(hfin : N.fg) (h : function.surjective (algebra_map R S)) : (submodule.restrict_scalars R N).fg | begin
obtain ⟨X, rfl⟩ := hfin,
use X,
exact (submodule.restrict_scalars_span R S h ↑X).symm
end | lemma | submodule.fg_restrict_scalars | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_group",
"algebra",
"algebra_map",
"comm_semiring",
"is_scalar_tower",
"module",
"semiring",
"submodule",
"submodule.restrict_scalars",
"submodule.restrict_scalars_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg.stablizes_of_supr_eq {M' : submodule R M} (hM' : M'.fg)
(N : ℕ →o submodule R M) (H : supr N = M') : ∃ n, M' = N n | begin
obtain ⟨S, hS⟩ := hM',
have : ∀ s : S, ∃ n, (s : M) ∈ N n :=
λ s, (submodule.mem_supr_of_chain N s).mp
(by { rw [H, ← hS], exact submodule.subset_span s.2 }),
choose f hf,
use S.attach.sup f,
apply le_antisymm,
{ conv_lhs { rw ← hS },
rw submodule.span_le,
intros s hs,
exact N.2 ... | lemma | submodule.fg.stablizes_of_supr_eq | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finset.le_sup",
"le_supr",
"submodule",
"submodule.mem_supr_of_chain",
"submodule.span_le",
"submodule.subset_span",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_iff_compact (s : submodule R M) : s.fg ↔ complete_lattice.is_compact_element s | begin
classical,
-- Introduce shorthand for span of an element
let sp : M → submodule R M := λ a, span R {a},
-- Trivial rewrite lemma; a small hack since simp (only) & rw can't accomplish this smoothly.
have supr_rw : ∀ t : finset M, (⨆ x ∈ t, sp x) = (⨆ x ∈ (↑t : set M), sp x), from λ t, by refl,
split,
... | theorem | submodule.fg_iff_compact | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"Sup_eq_supr",
"Sup_le_Sup",
"complete_lattice.finset_sup_compact_of_compact",
"complete_lattice.is_compact_element",
"finset",
"finset.sup_eq_supr",
"finset.sup_id_eq_Sup",
"finset.sup_image",
"submodule",
"supr_image"
] | Finitely generated submodules are precisely compact elements in the submodule lattice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fg.map₂ (f : M →ₗ[R] N →ₗ[R] P) {p : submodule R M} {q : submodule R N}
(hp : p.fg) (hq : q.fg) : (map₂ f p q).fg | let ⟨sm, hfm, hm⟩ := fg_def.1 hp, ⟨sn, hfn, hn⟩ := fg_def.1 hq in
fg_def.2 ⟨set.image2 (λ m n, f m n) sm sn,
hfm.image2 _ hfn, map₂_span_span R f sm sn ▸ hm ▸ hn ▸ rfl⟩ | theorem | submodule.fg.map₂ | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg.mul (hm : M.fg) (hn : N.fg) : (M * N).fg | hm.map₂ _ hn | theorem | submodule.fg.mul | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg.pow (h : M.fg) (n : ℕ) : (M ^ n).fg | nat.rec_on n
(⟨{1}, by simp [one_eq_span]⟩)
(λ n ih, by simpa [pow_succ] using h.mul ih) | lemma | submodule.fg.pow | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"ih",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg (I : ideal R) : Prop | ∃ S : finset R, ideal.span ↑S = I | def | ideal.fg | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finset",
"ideal",
"ideal.span"
] | An ideal of `R` is finitely generated if it is the span of a finite subset of `R`.
This is defeq to `submodule.fg`, but unfolds more nicely. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fg.map {R S : Type*} [semiring R] [semiring S] {I : ideal R} (h : I.fg)
(f : R →+* S) : (I.map f).fg | begin
classical,
obtain ⟨s, hs⟩ := h,
refine ⟨s.image f, _⟩,
rw [finset.coe_image, ←ideal.map_span, hs],
end | lemma | ideal.fg.map | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finset.coe_image",
"ideal",
"semiring"
] | The image of a finitely generated ideal is finitely generated.
This is the `ideal` version of `submodule.fg.map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fg_ker_comp {R S A : Type*} [comm_ring R] [comm_ring S] [comm_ring A]
(f : R →+* S) (g : S →+* A) (hf : f.ker.fg) (hg : g.ker.fg) (hsur : function.surjective f) :
(g.comp f).ker.fg | begin
letI : algebra R S := ring_hom.to_algebra f,
letI : algebra R A := ring_hom.to_algebra (g.comp f),
letI : algebra S A := ring_hom.to_algebra g,
letI : is_scalar_tower R S A := is_scalar_tower.of_algebra_map_eq (λ _, rfl),
let f₁ := algebra.linear_map R S,
let g₁ := (is_scalar_tower.to_alg_hom R S A).t... | lemma | ideal.fg_ker_comp | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"algebra",
"algebra.linear_map",
"comm_ring",
"is_scalar_tower",
"is_scalar_tower.of_algebra_map_eq",
"is_scalar_tower.to_alg_hom",
"ring_hom.to_algebra",
"submodule.fg_ker_comp",
"submodule.fg_restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_radical_pow_le_of_fg {R : Type*} [comm_semiring R] (I : ideal R) (h : I.radical.fg) :
∃ n : ℕ, I.radical ^ n ≤ I | begin
have := le_refl I.radical, revert this,
refine submodule.fg_induction _ _ (λ J, J ≤ I.radical → ∃ n : ℕ, J ^ n ≤ I) _ _ _ h,
{ intros x hx, obtain ⟨n, hn⟩ := hx (subset_span (set.mem_singleton x)),
exact ⟨n, by rwa [← ideal.span, span_singleton_pow, span_le, set.singleton_subset_iff]⟩ },
{ intros J K ... | lemma | ideal.exists_radical_pow_le_of_fg | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_pow",
"comm_semiring",
"finset.sup_le_iff",
"ideal",
"ideal.add_eq_sup",
"ideal.mem_sup_left",
"ideal.mem_sup_right",
"ideal.pow_le_pow",
"ideal.span",
"ideal.sum_eq_sup",
"set.mem_singleton",
"set.singleton_subset_iff",
"submodule.fg_induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.finite [semiring R] [add_comm_monoid M] [module R M] :
Prop | (out : (⊤ : submodule R M).fg) | class | module.finite | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"module",
"semiring",
"submodule"
] | A module over a semiring is `finite` if it is finitely generated as a module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_def {R M} [semiring R] [add_comm_monoid M] [module R M] :
finite R M ↔ (⊤ : submodule R M).fg | ⟨λ h, h.1, λ h, ⟨h⟩⟩ | lemma | module.finite_def | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"finite",
"module",
"semiring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_add_monoid_fg {M : Type*} [add_comm_monoid M] : module.finite ℕ M ↔ add_monoid.fg M | ⟨λ h, add_monoid.fg_def.2 $ (fg_iff_add_submonoid_fg ⊤).1 (finite_def.1 h),
λ h, finite_def.2 $ (fg_iff_add_submonoid_fg ⊤).2 (add_monoid.fg_def.1 h)⟩ | lemma | module.finite.iff_add_monoid_fg | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"add_monoid.fg",
"module.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_add_group_fg {G : Type*} [add_comm_group G] : module.finite ℤ G ↔ add_group.fg G | ⟨λ h, add_group.fg_def.2 $ (fg_iff_add_subgroup_fg ⊤).1 (finite_def.1 h),
λ h, finite_def.2 $ (fg_iff_add_subgroup_fg ⊤).2 (add_group.fg_def.1 h)⟩ | lemma | module.finite.iff_add_group_fg | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_group",
"add_group.fg",
"module.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_fin [finite R M] : ∃ (n : ℕ) (s : fin n → M), span R (range s) = ⊤ | submodule.fg_iff_exists_fin_generating_family.mp out | lemma | module.finite.exists_fin | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective [hM : finite R M] (f : M →ₗ[R] N) (hf : surjective f) :
finite R N | ⟨begin
rw [← linear_map.range_eq_top.2 hf, ← submodule.map_top],
exact hM.1.map f
end⟩ | lemma | module.finite.of_surjective | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"submodule.map_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range [finite R M] (f : M →ₗ[R] N) : finite R f.range | of_surjective f.range_restrict $ λ ⟨x, y, hy⟩, ⟨y, subtype.ext hy⟩ | instance | module.finite.range | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"subtype.ext"
] | The range of a linear map from a finite module is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (p : submodule R M) [finite R p] (f : M →ₗ[R] N) : finite R (p.map f) | of_surjective (f.restrict $ λ _, mem_map_of_mem) $ λ ⟨x, y, hy, hy'⟩, ⟨⟨_, hy⟩, subtype.ext hy'⟩ | instance | module.finite.map | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"submodule",
"subtype.ext"
] | Pushforwards of finite submodules are finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self : finite R R | ⟨⟨{1}, by simpa only [finset.coe_singleton] using ideal.span_singleton_one⟩⟩ | instance | module.finite.self | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"finset.coe_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_restrict_scalars_finite (R A M : Type*) [comm_semiring R] [semiring A] [add_comm_monoid M]
[module R M] [module A M] [algebra R A] [is_scalar_tower R A M] [hM : finite R M] :
finite A M | begin
rw [finite_def, fg_def] at hM ⊢,
obtain ⟨S, hSfin, hSgen⟩ := hM,
refine ⟨S, hSfin, eq_top_iff.2 _⟩,
have := submodule.span_le_restrict_scalars R A S,
rw hSgen at this,
exact this
end | lemma | module.finite.of_restrict_scalars_finite | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"algebra",
"comm_semiring",
"finite",
"is_scalar_tower",
"module",
"semiring",
"submodule.span_le_restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod [hM : finite R M] [hN : finite R N] : finite R (M × N) | ⟨begin
rw ← submodule.prod_top,
exact hM.1.prod hN.1
end⟩ | instance | module.finite.prod | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"submodule.prod_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi {ι : Type*} {M : ι → Type*} [_root_.finite ι] [Π i, add_comm_monoid (M i)]
[Π i, module R (M i)] [h : ∀ i, finite R (M i)] : finite R (Π i, M i) | ⟨begin
rw ← submodule.pi_top,
exact submodule.fg_pi (λ i, (h i).1),
end⟩ | instance | module.finite.pi | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"finite",
"module",
"submodule.fg_pi",
"submodule.pi_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv [hM : finite R M] (e : M ≃ₗ[R] N) : finite R N | of_surjective (e : M →ₗ[R] N) e.surjective | lemma | module.finite.equiv | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"equiv",
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans {R : Type*} (A M : Type*) [comm_semiring R] [semiring A] [algebra R A]
[add_comm_monoid M] [module R M] [module A M] [is_scalar_tower R A M] :
∀ [finite R A] [finite A M], finite R M | | ⟨⟨s, hs⟩⟩ ⟨⟨t, ht⟩⟩ := ⟨submodule.fg_def.2
⟨set.image2 (•) (↑s : set A) (↑t : set M),
set.finite.image2 _ s.finite_to_set t.finite_to_set,
by rw [set.image2_smul, submodule.span_smul_of_span_eq_top hs (↑t : set M),
ht, submodule.restrict_scalars_top]⟩⟩ | lemma | module.finite.trans | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"algebra",
"comm_semiring",
"finite",
"is_scalar_tower",
"module",
"semiring",
"set.finite.image2",
"set.image2_smul",
"submodule.restrict_scalars_top",
"submodule.span_smul_of_span_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.finite.base_change [comm_semiring R] [semiring A] [algebra R A]
[add_comm_monoid M] [module R M] [h : module.finite R M] :
module.finite A (tensor_product R A M) | begin
classical,
obtain ⟨s, hs⟩ := h.out,
refine ⟨⟨s.image (tensor_product.mk R A M 1), eq_top_iff.mpr $ λ x _, _⟩⟩,
apply tensor_product.induction_on x,
{ exact zero_mem _ },
{ intros x y,
rw [finset.coe_image, ← submodule.span_span_of_tower R, submodule.span_image, hs,
submodule.map_top, linear_... | instance | module.finite.base_change | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"algebra",
"comm_semiring",
"finset.coe_image",
"linear_map.range_coe",
"module",
"module.finite",
"mul_one",
"semiring",
"set.mem_range_self",
"set.range",
"smul_eq_mul",
"submodule.add_mem",
"submodule.map_top",
"submodule.smul_mem",
"submodule.span",
"submodule.... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.finite.tensor_product [comm_semiring R]
[add_comm_monoid M] [module R M] [add_comm_monoid N] [module R N]
[hM : module.finite R M] [hN : module.finite R N] : module.finite R (tensor_product R M N) | { out := (tensor_product.map₂_mk_top_top_eq_top R M N).subst (hM.out.map₂ _ hN.out) } | instance | module.finite.tensor_product | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"add_comm_monoid",
"comm_semiring",
"module",
"module.finite",
"tensor_product",
"tensor_product.map₂_mk_top_top_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite (f : A →+* B) : Prop | by letI : algebra A B := f.to_algebra; exact module.finite A B | def | ring_hom.finite | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"algebra",
"finite",
"module.finite"
] | A ring morphism `A →+* B` is `finite` if `B` is finitely generated as `A`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : finite (ring_hom.id A) | module.finite.self A | lemma | ring_hom.finite.id | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"module.finite.self",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective (f : A →+* B) (hf : surjective f) : f.finite | begin
letI := f.to_algebra,
exact module.finite.of_surjective (algebra.of_id A B).to_linear_map hf
end | lemma | ring_hom.finite.of_surjective | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"algebra.of_id",
"module.finite.of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : B →+* C} {f : A →+* B} (hg : g.finite) (hf : f.finite) : (g.comp f).finite | begin
letI := f.to_algebra,
letI := g.to_algebra,
letI := (g.comp f).to_algebra,
letI : is_scalar_tower A B C := restrict_scalars.is_scalar_tower A B C,
letI : module.finite A B := hf,
letI : module.finite B C := hg,
exact module.finite.trans B C,
end | lemma | ring_hom.finite.comp | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"is_scalar_tower",
"module.finite",
"module.finite.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_finite {f : A →+* B} {g : B →+* C} (h : (g.comp f).finite) : g.finite | begin
letI := f.to_algebra,
letI := g.to_algebra,
letI := (g.comp f).to_algebra,
letI : is_scalar_tower A B C := restrict_scalars.is_scalar_tower A B C,
letI : module.finite A C := h,
exact module.finite.of_restrict_scalars_finite A B C
end | lemma | ring_hom.finite.of_comp_finite | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"is_scalar_tower",
"module.finite",
"module.finite.of_restrict_scalars_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite (f : A →ₐ[R] B) : Prop | f.to_ring_hom.finite | def | alg_hom.finite | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite"
] | An algebra morphism `A →ₐ[R] B` is finite if it is finite as ring morphism.
In other words, if `B` is finitely generated as `A`-module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : finite (alg_hom.id R A) | ring_hom.finite.id A | lemma | alg_hom.finite.id | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"alg_hom.id",
"finite",
"ring_hom.finite.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.finite) (hf : f.finite) : (g.comp f).finite | ring_hom.finite.comp hg hf | lemma | alg_hom.finite.comp | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"ring_hom.finite.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective (f : A →ₐ[R] B) (hf : surjective f) : f.finite | ring_hom.finite.of_surjective f hf | lemma | alg_hom.finite.of_surjective | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"ring_hom.finite.of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_finite {f : A →ₐ[R] B} {g : B →ₐ[R] C} (h : (g.comp f).finite) : g.finite | ring_hom.finite.of_comp_finite h | lemma | alg_hom.finite.of_comp_finite | ring_theory | src/ring_theory/finiteness.lean | [
"algebra.algebra.restrict_scalars",
"algebra.algebra.subalgebra.basic",
"group_theory.finiteness",
"ring_theory.ideal.operations"
] | [
"finite",
"ring_hom.finite.of_comp_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.finite_presentation [comm_semiring R] [semiring A] [algebra R A] : Prop | ∃ (n : ℕ) (f : mv_polynomial (fin n) R →ₐ[R] A),
surjective f ∧ f.to_ring_hom.ker.fg | def | algebra.finite_presentation | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"algebra",
"comm_semiring",
"mv_polynomial",
"semiring"
] | An algebra over a commutative semiring is `finite_presentation` if it is the quotient of a
polynomial ring in `n` variables by a finitely generated ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_finite_presentation : finite_presentation R A → finite_type R A | begin
rintro ⟨n, f, hf⟩,
apply (finite_type.iff_quotient_mv_polynomial'').2,
exact ⟨n, f, hf.1⟩
end | lemma | algebra.finite_type.of_finite_presentation | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [] | A finitely presented algebra is of finite type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_finite_type [is_noetherian_ring R] : finite_type R A ↔ finite_presentation R A | begin
refine ⟨λ h, _, algebra.finite_type.of_finite_presentation⟩,
obtain ⟨n, f, hf⟩ := algebra.finite_type.iff_quotient_mv_polynomial''.1 h,
refine ⟨n, f, hf, _⟩,
have hnoet : is_noetherian_ring (mv_polynomial (fin n) R) := by apply_instance,
replace hnoet := (is_noetherian_ring_iff.1 hnoet).noetherian,
ex... | lemma | algebra.finite_presentation.of_finite_type | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"is_noetherian_ring",
"mv_polynomial"
] | An algebra over a Noetherian ring is finitely generated if and only if it is finitely
presented. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv (hfp : finite_presentation R A) (e : A ≃ₐ[R] B) : finite_presentation R B | begin
obtain ⟨n, f, hf⟩ := hfp,
use [n, alg_hom.comp ↑e f],
split,
{ exact function.surjective.comp e.surjective hf.1 },
suffices hker : (alg_hom.comp ↑e f).to_ring_hom.ker = f.to_ring_hom.ker,
{ rw hker, exact hf.2 },
{ have hco : (alg_hom.comp ↑e f).to_ring_hom = ring_hom.comp ↑e.to_ring_equiv f.to_ring... | lemma | algebra.finite_presentation.equiv | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"alg_hom.comp",
"equiv",
"ideal.comap_comap",
"ring_hom.comp",
"ring_hom.ker_coe_equiv",
"ring_hom.ker_eq_comap_bot"
] | If `e : A ≃ₐ[R] B` and `A` is finitely presented, then so is `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mv_polynomial (ι : Type u_2) [finite ι] :
finite_presentation R (mv_polynomial ι R) | by casesI nonempty_fintype ι; exact
let eqv := (mv_polynomial.rename_equiv R $ fintype.equiv_fin ι).symm in
⟨fintype.card ι, eqv, eqv.surjective,
((ring_hom.injective_iff_ker_eq_bot _).1 eqv.injective).symm ▸ submodule.fg_bot⟩ | lemma | algebra.finite_presentation.mv_polynomial | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"finite",
"fintype.equiv_fin",
"mv_polynomial",
"mv_polynomial.rename_equiv",
"nonempty_fintype",
"ring_hom.injective_iff_ker_eq_bot"
] | The ring of polynomials in finitely many variables is finitely presented. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self : finite_presentation R R | equiv (finite_presentation.mv_polynomial R pempty) (mv_polynomial.is_empty_alg_equiv R pempty) | lemma | algebra.finite_presentation.self | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"equiv",
"mv_polynomial.is_empty_alg_equiv",
"pempty"
] | `R` is finitely presented as `R`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial : finite_presentation R R[X] | equiv (finite_presentation.mv_polynomial R punit) (mv_polynomial.punit_alg_equiv R) | lemma | algebra.finite_presentation.polynomial | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"equiv",
"mv_polynomial.punit_alg_equiv",
"polynomial"
] | `R[X]` is finitely presented as `R`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient {I : ideal A} (h : I.fg) (hfp : finite_presentation R A) :
finite_presentation R (A ⧸ I) | begin
obtain ⟨n, f, hf⟩ := hfp,
refine ⟨n, (ideal.quotient.mkₐ R I).comp f, _, _⟩,
{ exact (ideal.quotient.mkₐ_surjective R I).comp hf.1 },
{ refine ideal.fg_ker_comp _ _ hf.2 _ hf.1,
simp [h] }
end | lemma | algebra.finite_presentation.quotient | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.fg_ker_comp",
"ideal.quotient.mkₐ",
"ideal.quotient.mkₐ_surjective"
] | The quotient of a finitely presented algebra by a finitely generated ideal is finitely
presented. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_surjective {f : A →ₐ[R] B} (hf : function.surjective f) (hker : f.to_ring_hom.ker.fg)
(hfp : finite_presentation R A) : finite_presentation R B | equiv (hfp.quotient hker) (ideal.quotient_ker_alg_equiv_of_surjective hf) | lemma | algebra.finite_presentation.of_surjective | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"equiv",
"ideal.quotient_ker_alg_equiv_of_surjective"
] | If `f : A →ₐ[R] B` is surjective with finitely generated kernel and `A` is finitely presented,
then so is `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iff : finite_presentation R A ↔
∃ n (I : ideal (mv_polynomial (fin n) R)) (e : (_ ⧸ I) ≃ₐ[R] A), I.fg | begin
split,
{ rintros ⟨n, f, hf⟩,
exact ⟨n, f.to_ring_hom.ker, ideal.quotient_ker_alg_equiv_of_surjective hf.1, hf.2⟩ },
{ rintros ⟨n, I, e, hfg⟩,
exact equiv ((finite_presentation.mv_polynomial R _).quotient hfg) e }
end | lemma | algebra.finite_presentation.iff | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"equiv",
"ideal",
"ideal.quotient_ker_alg_equiv_of_surjective",
"mv_polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_quotient_mv_polynomial' : finite_presentation R A ↔ ∃ (ι : Type u_2) (_ : fintype ι)
(f : mv_polynomial ι R →ₐ[R] A), surjective f ∧ f.to_ring_hom.ker.fg | begin
split,
{ rintro ⟨n, f, hfs, hfk⟩,
set ulift_var := mv_polynomial.rename_equiv R equiv.ulift,
refine ⟨ulift (fin n), infer_instance, f.comp ulift_var.to_alg_hom,
hfs.comp ulift_var.surjective,
ideal.fg_ker_comp _ _ _ hfk ulift_var.surjective⟩,
convert submodule.fg_bot,
exact ring_ho... | lemma | algebra.finite_presentation.iff_quotient_mv_polynomial' | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"alg_equiv.symm",
"equiv",
"equiv.symm",
"equiv.ulift",
"fintype",
"fintype.equiv_fin",
"ideal.fg_ker_comp",
"mv_polynomial",
"mv_polynomial.rename_equiv",
"ring_hom.ker_coe_equiv",
"submodule.fg_bot"
] | An algebra is finitely presented if and only if it is a quotient of a polynomial ring whose
variables are indexed by a fintype by a finitely generated ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mv_polynomial_of_finite_presentation (hfp : finite_presentation R A) (ι : Type*)
[finite ι] : finite_presentation R (mv_polynomial ι A) | begin
rw iff_quotient_mv_polynomial' at hfp ⊢,
classical,
obtain ⟨ι', _, f, hf_surj, hf_ker⟩ := hfp,
resetI,
let g := (mv_polynomial.map_alg_hom f).comp (mv_polynomial.sum_alg_equiv R ι ι').to_alg_hom,
casesI nonempty_fintype (ι ⊕ ι'),
refine ⟨ι ⊕ ι', by apply_instance, g,
(mv_polynomial.map_surjectiv... | lemma | algebra.finite_presentation.mv_polynomial_of_finite_presentation | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"alg_equiv.surjective",
"alg_hom.to_ring_hom_eq_coe",
"finite",
"ideal.fg_ker_comp",
"mv_polynomial",
"mv_polynomial.C",
"mv_polynomial.ker_map",
"mv_polynomial.map_alg_hom",
"mv_polynomial.map_alg_hom_coe_ring_hom",
"mv_polynomial.map_surjective",
"mv_polynomial.sum_alg_equiv",
"nonempty_fint... | If `A` is a finitely presented `R`-algebra, then `mv_polynomial (fin n) A` is finitely presented
as `R`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans [algebra A B] [is_scalar_tower R A B] (hfpA : finite_presentation R A)
(hfpB : finite_presentation A B) : finite_presentation R B | begin
obtain ⟨n, I, e, hfg⟩ := iff.1 hfpB,
exact equiv ((mv_polynomial_of_finite_presentation hfpA _).quotient hfg) (e.restrict_scalars R)
end | lemma | algebra.finite_presentation.trans | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"algebra",
"equiv",
"is_scalar_tower"
] | If `A` is an `R`-algebra and `S` is an `A`-algebra, both finitely presented, then `S` is
finitely presented as `R`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_restrict_scalars_finite_presentation [algebra A B] [is_scalar_tower R A B]
(hRB : finite_presentation R B) [hRA : finite_type R A] : finite_presentation A B | begin
classical,
obtain ⟨n, f, hf, s, hs⟩ := hRB,
let RX := mv_polynomial (fin n) R, let AX := mv_polynomial (fin n) A,
refine ⟨n, mv_polynomial.aeval (f ∘ X), _, _⟩,
{ rw [← algebra.range_top_iff_surjective, ← algebra.adjoin_range_eq_range_aeval, set.range_comp,
_root_.eq_top_iff, ← @adjoin_adjoin_of_t... | lemma | algebra.finite_presentation.of_restrict_scalars_finite_presentation | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"alg_hom.coe_to_ring_hom",
"alg_hom.to_ring_hom_eq_coe",
"algebra",
"algebra.adjoin",
"algebra.adjoin_range_eq_range_aeval",
"algebra.map_top",
"algebra.range_top_iff_surjective",
"algebra_map",
"finset.attach_eq_univ",
"finset.coe_image",
"finset.coe_union",
"finset.coe_univ",
"ideal.mul_me... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_fg_of_mv_polynomial {n : ℕ} (f : mv_polynomial (fin n) R →ₐ[R] A)
(hf : function.surjective f) (hfp : finite_presentation R A) : f.to_ring_hom.ker.fg | begin
classical,
obtain ⟨m, f', hf', s, hs⟩ := hfp,
let RXn := mv_polynomial (fin n) R, let RXm := mv_polynomial (fin m) R,
have := λ (i : fin n), hf' (f $ X i),
choose g hg,
have := λ (i : fin m), hf (f' $ X i),
choose h hh,
let aeval_h : RXm →ₐ[R] RXn := aeval h,
let g' : fin n → RXn := λ i, X i - a... | lemma | algebra.finite_presentation.ker_fg_of_mv_polynomial | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"alg_hom.coe_to_ring_hom",
"alg_hom.comp_algebra_map",
"alg_hom.to_ring_hom_eq_coe",
"finset.coe_image",
"finset.coe_union",
"finset.coe_univ",
"ideal.mul_mem_left",
"ideal.mul_mem_right",
"ideal.span",
"ideal.span_le",
"ideal.subset_span",
"map_mul",
"mv_polynomial",
"ring_hom.mem_ker",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_fg_of_surjective (f : A →ₐ[R] B) (hf : function.surjective f)
(hRA : finite_presentation R A) (hRB : finite_presentation R B) : f.to_ring_hom.ker.fg | begin
obtain ⟨n, g, hg, hg'⟩ := hRA,
convert (ker_fg_of_mv_polynomial (f.comp g) (hf.comp hg) hRB).map g.to_ring_hom,
simp_rw [ring_hom.ker_eq_comap_bot, alg_hom.to_ring_hom_eq_coe, alg_hom.comp_to_ring_hom],
rw [← ideal.comap_comap, ideal.map_comap_of_surjective (g : mv_polynomial (fin n) R →+* A) hg],
end | lemma | algebra.finite_presentation.ker_fg_of_surjective | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"alg_hom.comp_to_ring_hom",
"alg_hom.to_ring_hom_eq_coe",
"ideal.comap_comap",
"ideal.map_comap_of_surjective",
"mv_polynomial",
"ring_hom.ker_eq_comap_bot"
] | If `f : A →ₐ[R] B` is a sujection between finitely-presented `R`-algebras, then the kernel of
`f` is finitely generated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_presentation (f : A →+* B) : Prop | @algebra.finite_presentation A B _ _ f.to_algebra | def | ring_hom.finite_presentation | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"algebra.finite_presentation"
] | A ring morphism `A →+* B` is of `finite_presentation` if `B` is finitely presented as
`A`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_finite_presentation {f : A →+* B} (hf : f.finite_presentation) : f.finite_type | @algebra.finite_type.of_finite_presentation A B _ _ f.to_algebra hf | lemma | ring_hom.finite_type.of_finite_presentation | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"algebra.finite_type.of_finite_presentation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : finite_presentation (ring_hom.id A) | algebra.finite_presentation.self A | lemma | ring_hom.finite_presentation.id | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"algebra.finite_presentation.self",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_surjective {f : A →+* B} {g : B →+* C} (hf : f.finite_presentation) (hg : surjective g)
(hker : g.ker.fg) : (g.comp f).finite_presentation | @algebra.finite_presentation.of_surjective A B C _ _ f.to_algebra _ (g.comp f).to_algebra
{ to_fun := g, commutes' := λ a, rfl, .. g } hg hker hf | lemma | ring_hom.finite_presentation.comp_surjective | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"algebra.finite_presentation.of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective (f : A →+* B) (hf : surjective f) (hker : f.ker.fg) : f.finite_presentation | by { rw ← f.comp_id, exact (id A).comp_surjective hf hker} | lemma | ring_hom.finite_presentation.of_surjective | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_finite_type [is_noetherian_ring A] {f : A →+* B} : f.finite_type ↔ f.finite_presentation | @algebra.finite_presentation.of_finite_type A B _ _ f.to_algebra _ | lemma | ring_hom.finite_presentation.of_finite_type | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"algebra.finite_presentation.of_finite_type",
"is_noetherian_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : B →+* C} {f : A →+* B} (hg : g.finite_presentation) (hf : f.finite_presentation) :
(g.comp f).finite_presentation | @algebra.finite_presentation.trans A B C _ _ f.to_algebra _ (g.comp f).to_algebra g.to_algebra
{ smul_assoc := λ a b c, begin
simp only [algebra.smul_def, ring_hom.map_mul, mul_assoc],
refl
end }
hf hg | lemma | ring_hom.finite_presentation.comp | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"algebra.finite_presentation.trans",
"algebra.smul_def",
"mul_assoc",
"ring_hom.map_mul",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_finite_type (f : A →+* B) {g : B →+* C} (hg : (g.comp f).finite_presentation)
(hf : f.finite_type) : g.finite_presentation | @@algebra.finite_presentation.of_restrict_scalars_finite_presentation _ _ f.to_algebra _
(g.comp f).to_algebra g.to_algebra
(@@is_scalar_tower.of_algebra_map_eq' _ _ _ f.to_algebra g.to_algebra (g.comp f).to_algebra rfl)
hg hf | lemma | ring_hom.finite_presentation.of_comp_finite_type | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"algebra.finite_presentation.of_restrict_scalars_finite_presentation",
"is_scalar_tower.of_algebra_map_eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_presentation (f : A →ₐ[R] B) : Prop | f.to_ring_hom.finite_presentation | def | alg_hom.finite_presentation | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [] | An algebra morphism `A →ₐ[R] B` is of `finite_presentation` if it is of finite presentation as
ring morphism. In other words, if `B` is finitely presented as `A`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_finite_presentation {f : A →ₐ[R] B} (hf : f.finite_presentation) : f.finite_type | ring_hom.finite_type.of_finite_presentation hf | lemma | alg_hom.finite_type.of_finite_presentation | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"ring_hom.finite_type.of_finite_presentation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : finite_presentation (alg_hom.id R A) | ring_hom.finite_presentation.id A | lemma | alg_hom.finite_presentation.id | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"alg_hom.id",
"ring_hom.finite_presentation.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.finite_presentation)
(hf : f.finite_presentation) : (g.comp f).finite_presentation | ring_hom.finite_presentation.comp hg hf | lemma | alg_hom.finite_presentation.comp | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"ring_hom.finite_presentation.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_surjective {f : A →ₐ[R] B} {g : B →ₐ[R] C} (hf : f.finite_presentation)
(hg : surjective g) (hker : g.to_ring_hom.ker.fg) : (g.comp f).finite_presentation | ring_hom.finite_presentation.comp_surjective hf hg hker | lemma | alg_hom.finite_presentation.comp_surjective | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"ring_hom.finite_presentation.comp_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective (f : A →ₐ[R] B) (hf : surjective f) (hker : f.to_ring_hom.ker.fg) :
f.finite_presentation | ring_hom.finite_presentation.of_surjective f hf hker | lemma | alg_hom.finite_presentation.of_surjective | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"ring_hom.finite_presentation.of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_finite_type [is_noetherian_ring A] {f : A →ₐ[R] B} :
f.finite_type ↔ f.finite_presentation | ring_hom.finite_presentation.of_finite_type | lemma | alg_hom.finite_presentation.of_finite_type | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [
"is_noetherian_ring",
"ring_hom.finite_presentation.of_finite_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_finite_type (f : A →ₐ[R] B) {g : B →ₐ[R] C} (h : (g.comp f).finite_presentation)
(h' : f.finite_type) : g.finite_presentation | h.of_comp_finite_type _ h' | lemma | alg_hom.finite_presentation.of_comp_finite_type | ring_theory | src/ring_theory/finite_presentation.lean | [
"ring_theory.finite_type",
"ring_theory.mv_polynomial.tower",
"ring_theory.ideal.quotient_operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.finite_type [comm_semiring R] [semiring A] [algebra R A] : Prop | (out : (⊤ : subalgebra R A).fg) | class | algebra.finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra",
"comm_semiring",
"semiring",
"subalgebra"
] | An algebra over a commutative semiring is of `finite_type` if it is finitely generated
over the base ring as algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_type {R : Type*} (A : Type*) [comm_semiring R] [semiring A]
[algebra R A] [hRA : finite R A] : algebra.finite_type R A | ⟨subalgebra.fg_of_submodule_fg hRA.1⟩ | instance | module.finite.finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra",
"algebra.finite_type",
"comm_semiring",
"finite",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self : finite_type R R | ⟨⟨{1}, subsingleton.elim _ _⟩⟩ | lemma | algebra.finite_type.self | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial : finite_type R R[X] | ⟨⟨{polynomial.X}, by { rw finset.coe_singleton, exact polynomial.adjoin_X }⟩⟩ | lemma | algebra.finite_type.polynomial | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"finset.coe_singleton",
"polynomial",
"polynomial.X",
"polynomial.adjoin_X"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mv_polynomial (ι : Type*) [finite ι] : finite_type R (mv_polynomial ι R) | by casesI nonempty_fintype ι; exact ⟨⟨finset.univ.image mv_polynomial.X,
by {rw [finset.coe_image, finset.coe_univ, set.image_univ], exact mv_polynomial.adjoin_range_X}⟩⟩ | lemma | algebra.finite_type.mv_polynomial | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"finite",
"finset.coe_image",
"finset.coe_univ",
"mv_polynomial",
"mv_polynomial.X",
"mv_polynomial.adjoin_range_X",
"nonempty_fintype",
"set.image_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_restrict_scalars_finite_type [algebra A B] [is_scalar_tower R A B] [hB : finite_type R B] :
finite_type A B | begin
obtain ⟨S, hS⟩ := hB.out,
refine ⟨⟨S, eq_top_iff.2 (λ b, _)⟩⟩,
have le : adjoin R (S : set B) ≤ subalgebra.restrict_scalars R (adjoin A S),
{ apply (algebra.adjoin_le _ : _ ≤ (subalgebra.restrict_scalars R (adjoin A ↑S))),
simp only [subalgebra.coe_restrict_scalars],
exact algebra.subset_adjoin, }... | lemma | algebra.finite_type.of_restrict_scalars_finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra",
"algebra.adjoin_le",
"algebra.subset_adjoin",
"is_scalar_tower",
"subalgebra.coe_restrict_scalars",
"subalgebra.restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective (hRA : finite_type R A) (f : A →ₐ[R] B) (hf : surjective f) :
finite_type R B | ⟨begin
convert hRA.1.map f,
simpa only [map_top f, @eq_comm _ ⊤, eq_top_iff, alg_hom.mem_range] using hf
end⟩ | lemma | algebra.finite_type.of_surjective | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"alg_hom.mem_range",
"eq_top_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv (hRA : finite_type R A) (e : A ≃ₐ[R] B) : finite_type R B | hRA.of_surjective e e.surjective | lemma | algebra.finite_type.equiv | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans [algebra A B] [is_scalar_tower R A B] (hRA : finite_type R A) (hAB : finite_type A B) :
finite_type R B | ⟨fg_trans' hRA.1 hAB.1⟩ | lemma | algebra.finite_type.trans | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_quotient_mv_polynomial : (finite_type R A) ↔ ∃ (s : finset A)
(f : (mv_polynomial {x // x ∈ s} R) →ₐ[R] A), (surjective f) | begin
split,
{ rintro ⟨s, hs⟩,
use [s, mv_polynomial.aeval coe],
intro x,
have hrw : (↑s : set A) = (λ (x : A), x ∈ s.val) := rfl,
rw [← set.mem_range, ← alg_hom.coe_range, ← adjoin_eq_range, ← hrw, hs],
exact set.mem_univ x },
{ rintro ⟨s, ⟨f, hsur⟩⟩,
exact finite_type.of_surjective (fini... | lemma | algebra.finite_type.iff_quotient_mv_polynomial | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"alg_hom.coe_range",
"finset",
"mv_polynomial",
"mv_polynomial.aeval",
"set.mem_range",
"set.mem_univ"
] | An algebra is finitely generated if and only if it is a quotient
of a polynomial ring whose variables are indexed by a finset. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iff_quotient_mv_polynomial' : (finite_type R A) ↔ ∃ (ι : Type u_2) (_ : fintype ι)
(f : (mv_polynomial ι R) →ₐ[R] A), (surjective f) | begin
split,
{ rw iff_quotient_mv_polynomial,
rintro ⟨s, ⟨f, hsur⟩⟩,
use [{x // x ∈ s}, by apply_instance, f, hsur] },
{ rintro ⟨ι, ⟨hfintype, ⟨f, hsur⟩⟩⟩,
letI : fintype ι := hfintype,
exact finite_type.of_surjective (finite_type.mv_polynomial R ι) f hsur }
end | lemma | algebra.finite_type.iff_quotient_mv_polynomial' | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"fintype",
"mv_polynomial"
] | An algebra is finitely generated if and only if it is a quotient
of a polynomial ring whose variables are indexed by a fintype. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iff_quotient_mv_polynomial'' : (finite_type R A) ↔ ∃ (n : ℕ)
(f : (mv_polynomial (fin n) R) →ₐ[R] A), (surjective f) | begin
split,
{ rw iff_quotient_mv_polynomial',
rintro ⟨ι, hfintype, ⟨f, hsur⟩⟩,
resetI,
have equiv := mv_polynomial.rename_equiv R (fintype.equiv_fin ι),
exact ⟨fintype.card ι, alg_hom.comp f equiv.symm, function.surjective.comp hsur
(alg_equiv.symm equiv).surjective⟩ },
{ rintro ⟨n, ⟨f, hsu... | lemma | algebra.finite_type.iff_quotient_mv_polynomial'' | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"alg_equiv.symm",
"alg_hom.comp",
"equiv",
"equiv.symm",
"fintype.equiv_fin",
"mv_polynomial",
"mv_polynomial.rename_equiv"
] | An algebra is finitely generated if and only if it is a quotient of a polynomial ring in `n`
variables. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod [hA : finite_type R A] [hB : finite_type R B] : finite_type R (A × B) | ⟨begin
rw ← subalgebra.prod_top,
exact hA.1.prod hB.1
end⟩ | instance | algebra.finite_type.prod | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"subalgebra.prod_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring (R S : Type*) [comm_ring R] [comm_ring S] [algebra R S]
[h : algebra.finite_type R S] [is_noetherian_ring R] : is_noetherian_ring S | begin
obtain ⟨s, hs⟩ := h.1,
apply is_noetherian_ring_of_surjective
(mv_polynomial s R) S (mv_polynomial.aeval coe : mv_polynomial s R →ₐ[R] S),
rw [← set.range_iff_surjective, alg_hom.coe_to_ring_hom, ← alg_hom.coe_range,
← algebra.adjoin_range_eq_range_aeval, subtype.range_coe_subtype, finset.set_of_mem... | lemma | algebra.finite_type.is_noetherian_ring | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"alg_hom.coe_range",
"alg_hom.coe_to_ring_hom",
"algebra",
"algebra.adjoin_range_eq_range_aeval",
"algebra.finite_type",
"comm_ring",
"finset.set_of_mem",
"is_noetherian_ring",
"is_noetherian_ring_of_surjective",
"mv_polynomial",
"mv_polynomial.aeval",
"set.range_iff_surjective",
"subtype.ra... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.subalgebra.fg_iff_finite_type {R A : Type*} [comm_semiring R] [semiring A]
[algebra R A] (S : subalgebra R A) : S.fg ↔ algebra.finite_type R S | S.fg_top.symm.trans ⟨λ h, ⟨h⟩, λ h, h.out⟩ | lemma | subalgebra.fg_iff_finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra",
"algebra.finite_type",
"comm_semiring",
"semiring",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_type (f : A →+* B) : Prop | @algebra.finite_type A B _ _ f.to_algebra | def | ring_hom.finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra.finite_type"
] | A ring morphism `A →+* B` is of `finite_type` if `B` is finitely generated as `A`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_type {f : A →+* B} (hf : f.finite) : finite_type f | @module.finite.finite_type _ _ _ _ f.to_algebra hf | lemma | ring_hom.finite.finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"module.finite.finite_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : finite_type (ring_hom.id A) | algebra.finite_type.self A | lemma | ring_hom.finite_type.id | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra.finite_type.self",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_surjective {f : A →+* B} {g : B →+* C} (hf : f.finite_type) (hg : surjective g) :
(g.comp f).finite_type | @algebra.finite_type.of_surjective A B C _ _ f.to_algebra _ (g.comp f).to_algebra hf
{ to_fun := g, commutes' := λ a, rfl, .. g } hg | lemma | ring_hom.finite_type.comp_surjective | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra.finite_type.of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective (f : A →+* B) (hf : surjective f) : f.finite_type | by { rw ← f.comp_id, exact (id A).comp_surjective hf } | lemma | ring_hom.finite_type.of_surjective | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : B →+* C} {f : A →+* B} (hg : g.finite_type) (hf : f.finite_type) :
(g.comp f).finite_type | @algebra.finite_type.trans A B C _ _ f.to_algebra _ (g.comp f).to_algebra g.to_algebra
begin
fconstructor,
intros a b c,
simp only [algebra.smul_def, ring_hom.map_mul, mul_assoc],
refl
end
hf hg | lemma | ring_hom.finite_type.comp | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra.finite_type.trans",
"algebra.smul_def",
"mul_assoc",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_finite {f : A →+* B} (hf : f.finite) : f.finite_type | @module.finite.finite_type _ _ _ _ f.to_algebra hf | lemma | ring_hom.finite_type.of_finite | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"module.finite.finite_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_finite_type {f : A →+* B} {g : B →+* C} (h : (g.comp f).finite_type) :
g.finite_type | begin
letI := f.to_algebra,
letI := g.to_algebra,
letI := (g.comp f).to_algebra,
letI : is_scalar_tower A B C := restrict_scalars.is_scalar_tower A B C,
letI : algebra.finite_type A C := h,
exact algebra.finite_type.of_restrict_scalars_finite_type A B C
end | lemma | ring_hom.finite_type.of_comp_finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra.finite_type",
"algebra.finite_type.of_restrict_scalars_finite_type",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_type (f : A →ₐ[R] B) : Prop | f.to_ring_hom.finite_type | def | alg_hom.finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [] | An algebra morphism `A →ₐ[R] B` is of `finite_type` if it is of finite type as ring morphism.
In other words, if `B` is finitely generated as `A`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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