statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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derivation_to_square_zero_equiv_lift :
derivation R A I ≃
{ f : A →ₐ[R] B // (ideal.quotient.mkₐ R I).comp f = is_scalar_tower.to_alg_hom R A (B ⧸ I) } | begin
refine ⟨λ d, ⟨lift_of_derivation_to_square_zero I hI d, _⟩, λ f,
(derivation_to_square_zero_of_lift I hI f.1 f.2 : _), _, _⟩,
{ ext x, exact lift_of_derivation_to_square_zero_mk_apply I hI d x },
{ intro d, ext x, exact add_sub_cancel (d x : B) (algebra_map A B x) },
{ rintro ⟨f, hf⟩, ext x, exact su... | def | derivation_to_square_zero_equiv_lift | ring_theory.derivation | src/ring_theory/derivation/to_square_zero.lean | [
"ring_theory.derivation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"algebra_map",
"derivation",
"derivation_to_square_zero_of_lift",
"ideal.quotient.mkₐ",
"is_scalar_tower.to_alg_hom",
"lift_of_derivation_to_square_zero_mk_apply"
] | Given a tower of algebras `R → A → B`, and a square-zero `I : ideal B`,
there is a 1-1 correspondance between `R`-derivations from `A` to `I` and
lifts `A →ₐ[R] B` of the canonical map `A →ₐ[R] B ⧸ I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_valuation_ring (R : Type u) [comm_ring R] [is_domain R]
extends is_principal_ideal_ring R, local_ring R : Prop | (not_a_field' : maximal_ideal R ≠ ⊥) | class | discrete_valuation_ring | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"comm_ring",
"is_domain",
"is_principal_ideal_ring",
"local_ring"
] | An integral domain is a *discrete valuation ring* (DVR) if it's a local PID which
is not a field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_a_field : maximal_ideal R ≠ ⊥ | not_a_field' | lemma | discrete_valuation_ring.not_a_field | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_is_field : ¬ is_field R | local_ring.is_field_iff_maximal_ideal_eq.not.mpr (not_a_field R) | lemma | discrete_valuation_ring.not_is_field | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"is_field"
] | A discrete valuation ring `R` is not a field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_of_span_eq_maximal_ideal {R : Type*} [comm_ring R] [local_ring R] [is_domain R]
(ϖ : R) (hϖ : ϖ ≠ 0) (h : maximal_ideal R = ideal.span {ϖ}) : irreducible ϖ | begin
have h2 : ¬(is_unit ϖ) := show ϖ ∈ maximal_ideal R,
from h.symm ▸ submodule.mem_span_singleton_self ϖ,
refine ⟨h2, _⟩,
intros a b hab,
by_contra' h,
obtain ⟨ha : a ∈ maximal_ideal R, hb : b ∈ maximal_ideal R⟩ := h,
rw [h, mem_span_singleton'] at ha hb,
rcases ha with ⟨a, rfl⟩,
rcases hb with ⟨... | theorem | discrete_valuation_ring.irreducible_of_span_eq_maximal_ideal | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"comm_ring",
"eq_zero_of_mul_eq_self_right",
"ideal.span",
"irreducible",
"is_domain",
"is_unit",
"is_unit_of_dvd_one",
"local_ring",
"ring",
"submodule.mem_span_singleton_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_iff_uniformizer (ϖ : R) :
irreducible ϖ ↔ maximal_ideal R = ideal.span {ϖ} | ⟨λ hϖ, (eq_maximal_ideal (is_maximal_of_irreducible hϖ)).symm, λ h,
irreducible_of_span_eq_maximal_ideal ϖ (λ e, not_a_field R $ by rwa [h, span_singleton_eq_bot]) h⟩ | theorem | discrete_valuation_ring.irreducible_iff_uniformizer | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"ideal.span",
"irreducible"
] | An element of a DVR is irreducible iff it is a uniformizer, that is, generates the
maximal ideal of R | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.irreducible.maximal_ideal_eq {ϖ : R} (h : irreducible ϖ) :
maximal_ideal R = ideal.span {ϖ} | (irreducible_iff_uniformizer _).mp h | lemma | irreducible.maximal_ideal_eq | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"ideal.span",
"irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_irreducible : ∃ ϖ : R, irreducible ϖ | by {simp_rw [irreducible_iff_uniformizer],
exact (is_principal_ideal_ring.principal $ maximal_ideal R).principal} | theorem | discrete_valuation_ring.exists_irreducible | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"irreducible"
] | Uniformisers exist in a DVR | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_prime : ∃ ϖ : R, prime ϖ | (exists_irreducible R).imp (λ _, principal_ideal_ring.irreducible_iff_prime.1) | theorem | discrete_valuation_ring.exists_prime | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"prime"
] | Uniformisers exist in a DVR | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iff_pid_with_one_nonzero_prime (R : Type u) [comm_ring R] [is_domain R] :
discrete_valuation_ring R ↔ is_principal_ideal_ring R ∧ ∃! P : ideal R, P ≠ ⊥ ∧ is_prime P | begin
split,
{ intro RDVR,
rcases id RDVR with ⟨Rlocal⟩,
split, assumption,
resetI,
use local_ring.maximal_ideal R,
split, split,
{ assumption },
{ apply_instance } ,
{ rintro Q ⟨hQ1, hQ2⟩,
obtain ⟨q, rfl⟩ := (is_principal_ideal_ring.principal Q).1,
have hq : q ≠ 0,
... | theorem | discrete_valuation_ring.iff_pid_with_one_nonzero_prime | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"comm_ring",
"discrete_valuation_ring",
"ideal",
"is_domain",
"is_principal_ideal_ring",
"le_bot_iff",
"local_ring",
"local_ring.maximal_ideal",
"local_ring.of_unique_nonzero_prime"
] | an integral domain is a DVR iff it's a PID with a unique non-zero prime ideal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_of_irreducible {a b : R} (ha : irreducible a) (hb : irreducible b) :
associated a b | begin
rw irreducible_iff_uniformizer at ha hb,
rw [←span_singleton_eq_span_singleton, ←ha, hb],
end | lemma | discrete_valuation_ring.associated_of_irreducible | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"associated",
"irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_unit_mul_pow_irreducible_factorization [comm_ring R] : Prop | ∃ p : R, irreducible p ∧ ∀ {x : R}, x ≠ 0 → ∃ (n : ℕ), associated (p ^ n) x | def | discrete_valuation_ring.has_unit_mul_pow_irreducible_factorization | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"associated",
"comm_ring",
"irreducible"
] | Alternative characterisation of discrete valuation rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_irreducible ⦃p q : R⦄ (hp : irreducible p) (hq : irreducible q) :
associated p q | begin
rcases hR with ⟨ϖ, hϖ, hR⟩,
suffices : ∀ {p : R} (hp : irreducible p), associated p ϖ,
{ apply associated.trans (this hp) (this hq).symm, },
clear hp hq p q,
intros p hp,
obtain ⟨n, hn⟩ := hR hp.ne_zero,
have : irreducible (ϖ ^ n) := hn.symm.irreducible hp,
rcases lt_trichotomy n 1 with (H|rfl|H),... | lemma | discrete_valuation_ring.has_unit_mul_pow_irreducible_factorization.unique_irreducible | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"associated",
"associated.trans",
"irreducible",
"is_unit_of_mul_is_unit_left",
"nat.exists_eq_add_of_lt",
"not_irreducible_one",
"pow_one",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_unique_factorization_monoid : unique_factorization_monoid R | let p := classical.some hR in
let spec := classical.some_spec hR in
unique_factorization_monoid.of_exists_prime_factors $ λ x hx,
begin
use multiset.replicate (classical.some (spec.2 hx)) p,
split,
{ intros q hq,
have hpq := multiset.eq_of_mem_replicate hq,
rw hpq,
refine ⟨spec.1.ne_zero, spec.1.not_u... | theorem | discrete_valuation_ring.has_unit_mul_pow_irreducible_factorization.to_unique_factorization_monoid | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"dvd_mul_of_dvd_left",
"dvd_rfl",
"dvd_zero",
"is_unit.dvd_mul_left",
"is_unit.dvd_mul_right",
"mul_assoc",
"mul_left_comm",
"multiset.eq_of_mem_replicate",
"multiset.prod_replicate",
"multiset.replicate",
"one_mul",
"pow_succ",
"pow_zero",
"unique_factorization_monoid",
"unique_factoriz... | An integral domain in which there is an irreducible element `p`
such that every nonzero element is associated to a power of `p` is a unique factorization domain.
See `discrete_valuation_ring.of_has_unit_mul_pow_irreducible_factorization`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_ufd_of_unique_irreducible [unique_factorization_monoid R]
(h₁ : ∃ p : R, irreducible p)
(h₂ : ∀ ⦃p q : R⦄, irreducible p → irreducible q → associated p q) :
has_unit_mul_pow_irreducible_factorization R | begin
obtain ⟨p, hp⟩ := h₁,
refine ⟨p, hp, _⟩,
intros x hx,
cases wf_dvd_monoid.exists_factors x hx with fx hfx,
refine ⟨fx.card, _⟩,
have H := hfx.2,
rw ← associates.mk_eq_mk_iff_associated at H ⊢,
rw [← H, ← associates.prod_mk, associates.mk_pow, ← multiset.prod_replicate],
congr' 1,
symmetry,
r... | lemma | discrete_valuation_ring.has_unit_mul_pow_irreducible_factorization.of_ufd_of_unique_irreducible | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"and_imp",
"associated",
"associates.mk_eq_mk_iff_associated",
"associates.mk_pow",
"associates.prod_mk",
"exists_imp_distrib",
"irreducible",
"multiset.card_map",
"multiset.eq_replicate",
"multiset.mem_map",
"multiset.prod_replicate",
"unique_factorization_monoid",
"wf_dvd_monoid.exists_fac... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aux_pid_of_ufd_of_unique_irreducible
(R : Type u) [comm_ring R] [is_domain R] [unique_factorization_monoid R]
(h₁ : ∃ p : R, irreducible p)
(h₂ : ∀ ⦃p q : R⦄, irreducible p → irreducible q → associated p q) :
is_principal_ideal_ring R | begin
constructor,
intro I,
by_cases I0 : I = ⊥, { rw I0, use 0, simp only [set.singleton_zero, submodule.span_zero], },
obtain ⟨x, hxI, hx0⟩ : ∃ x ∈ I, x ≠ (0:R) := I.ne_bot_iff.mp I0,
obtain ⟨p, hp, H⟩ :=
has_unit_mul_pow_irreducible_factorization.of_ufd_of_unique_irreducible h₁ h₂,
have ex : ∃ n : ℕ,... | lemma | discrete_valuation_ring.aux_pid_of_ufd_of_unique_irreducible | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"associated",
"comm_ring",
"ideal.span",
"irreducible",
"is_domain",
"is_principal_ideal_ring",
"is_unit.dvd_mul_right",
"pow_dvd_pow",
"submodule.span_singleton_le_iff_mem",
"submodule.span_zero",
"submodule.zero_mem",
"unique_factorization_monoid",
"units.is_unit",
"units.mul_inv_cancel_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_ufd_of_unique_irreducible
{R : Type u} [comm_ring R] [is_domain R] [unique_factorization_monoid R]
(h₁ : ∃ p : R, irreducible p)
(h₂ : ∀ ⦃p q : R⦄, irreducible p → irreducible q → associated p q) :
discrete_valuation_ring R | begin
rw iff_pid_with_one_nonzero_prime,
haveI PID : is_principal_ideal_ring R := aux_pid_of_ufd_of_unique_irreducible R h₁ h₂,
obtain ⟨p, hp⟩ := h₁,
refine ⟨PID, ⟨ideal.span {p}, ⟨_, _⟩, _⟩⟩,
{ rw submodule.ne_bot_iff,
refine ⟨p, ideal.mem_span_singleton.mpr (dvd_refl p), hp.ne_zero⟩, },
{ rwa [ideal.s... | lemma | discrete_valuation_ring.of_ufd_of_unique_irreducible | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"associated",
"comm_ring",
"discrete_valuation_ring",
"dvd_refl",
"ideal.span_singleton_prime",
"irreducible",
"is_domain",
"is_principal_ideal_ring",
"submodule.is_principal.span_singleton_generator",
"submodule.ne_bot_iff",
"unique_factorization_monoid"
] | A unique factorization domain with at least one irreducible element
in which all irreducible elements are associated
is a discrete valuation ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_has_unit_mul_pow_irreducible_factorization {R : Type u} [comm_ring R] [is_domain R]
(hR : has_unit_mul_pow_irreducible_factorization R) :
discrete_valuation_ring R | begin
letI : unique_factorization_monoid R := hR.to_unique_factorization_monoid,
apply of_ufd_of_unique_irreducible _ hR.unique_irreducible,
unfreezingI { obtain ⟨p, hp, H⟩ := hR, exact ⟨p, hp⟩, },
end | lemma | discrete_valuation_ring.of_has_unit_mul_pow_irreducible_factorization | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"comm_ring",
"discrete_valuation_ring",
"is_domain",
"unique_factorization_monoid"
] | An integral domain in which there is an irreducible element `p`
such that every nonzero element is associated to a power of `p`
is a discrete valuation ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associated_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : irreducible ϖ) :
∃ (n : ℕ), associated x (ϖ ^ n) | begin
have : wf_dvd_monoid R := is_noetherian_ring.wf_dvd_monoid,
cases wf_dvd_monoid.exists_factors x hx with fx hfx,
unfreezingI { use fx.card },
have H := hfx.2,
rw ← associates.mk_eq_mk_iff_associated at H ⊢,
rw [← H, ← associates.prod_mk, associates.mk_pow, ← multiset.prod_replicate],
congr' 1,
rw ... | lemma | discrete_valuation_ring.associated_pow_irreducible | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"and_imp",
"associated",
"associates.mk_eq_mk_iff_associated",
"associates.mk_pow",
"associates.prod_mk",
"exists_imp_distrib",
"irreducible",
"is_noetherian_ring.wf_dvd_monoid",
"multiset.card_map",
"multiset.eq_replicate",
"multiset.mem_map",
"multiset.prod_replicate",
"wf_dvd_monoid",
"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_unit_mul_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : irreducible ϖ) :
∃ (n : ℕ) (u : Rˣ), x = u * ϖ ^ n | begin
obtain ⟨n, hn⟩ := associated_pow_irreducible hx hirr,
obtain ⟨u, rfl⟩ := hn.symm,
use [n, u],
apply mul_comm,
end | lemma | discrete_valuation_ring.eq_unit_mul_pow_irreducible | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"irreducible",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_eq_span_pow_irreducible {s : ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : irreducible ϖ) :
∃ n : ℕ, s = ideal.span {ϖ ^ n} | begin
have gen_ne_zero : generator s ≠ 0,
{ rw [ne.def, ← eq_bot_iff_generator_eq_zero], assumption },
rcases associated_pow_irreducible gen_ne_zero hirr with ⟨n, u, hnu⟩,
use n,
have : span _ = _ := span_singleton_generator s,
rw [← this, ← hnu, span_singleton_eq_span_singleton],
use u
end | lemma | discrete_valuation_ring.ideal_eq_span_pow_irreducible | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"ideal",
"ideal.span",
"irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_mul_pow_congr_pow {p q : R} (hp : irreducible p) (hq : irreducible q)
(u v : Rˣ) (m n : ℕ) (h : ↑u * p ^ m = v * q ^ n) :
m = n | begin
have key : associated (multiset.replicate m p).prod (multiset.replicate n q).prod,
{ rw [multiset.prod_replicate, multiset.prod_replicate, associated],
refine ⟨u * v⁻¹, _⟩,
simp only [units.coe_mul],
rw [mul_left_comm, ← mul_assoc, h, mul_right_comm, units.mul_inv, one_mul], },
have := multiset.... | lemma | discrete_valuation_ring.unit_mul_pow_congr_pow | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"associated",
"irreducible",
"mul_assoc",
"mul_left_comm",
"mul_right_comm",
"multiset.card_eq_card_of_rel",
"multiset.card_replicate",
"multiset.eq_of_mem_replicate",
"multiset.prod_replicate",
"multiset.replicate",
"one_mul",
"unique_factorization_monoid.factors_unique",
"units.coe_mul",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_mul_pow_congr_unit {ϖ : R} (hirr : irreducible ϖ) (u v : Rˣ) (m n : ℕ)
(h : ↑u * ϖ ^ m = v * ϖ ^ n) :
u = v | begin
obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h,
rw ← sub_eq_zero at h,
rw [← sub_mul, mul_eq_zero] at h,
cases h,
{ rw sub_eq_zero at h, exact_mod_cast h },
{ apply (hirr.ne_zero (pow_eq_zero h)).elim, }
end | lemma | discrete_valuation_ring.unit_mul_pow_congr_unit | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"irreducible",
"mul_eq_zero",
"pow_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val
(R : Type u) [comm_ring R] [is_domain R] [discrete_valuation_ring R] :
add_valuation R part_enat | add_valuation (classical.some_spec (exists_prime R)) | def | discrete_valuation_ring.add_val | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"add_valuation",
"comm_ring",
"discrete_valuation_ring",
"is_domain",
"part_enat"
] | The `part_enat`-valued additive valuation on a DVR | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_val_def (r : R) (u : Rˣ) {ϖ : R} (hϖ : irreducible ϖ) (n : ℕ) (hr : r = u * ϖ ^ n) :
add_val R r = n | by rw [add_val, add_valuation_apply, hr,
eq_of_associated_left (associated_of_irreducible R hϖ
(classical.some_spec (exists_prime R)).irreducible),
eq_of_associated_right (associated.symm ⟨u, mul_comm _ _⟩),
multiplicity_pow_self_of_prime (principal_ideal_ring.irreducible_iff_prime.1 hϖ)] | lemma | discrete_valuation_ring.add_val_def | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"associated.symm",
"irreducible",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_def' (u : Rˣ) {ϖ : R} (hϖ : irreducible ϖ) (n : ℕ) :
add_val R ((u : R) * ϖ ^ n) = n | add_val_def _ u hϖ n rfl | lemma | discrete_valuation_ring.add_val_def' | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_zero : add_val R 0 = ⊤ | (add_val R).map_zero | lemma | discrete_valuation_ring.add_val_zero | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_one : add_val R 1 = 0 | (add_val R).map_one | lemma | discrete_valuation_ring.add_val_one | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_uniformizer {ϖ : R} (hϖ : irreducible ϖ) : add_val R ϖ = 1 | by simpa only [one_mul, eq_self_iff_true, units.coe_one, pow_one, forall_true_left, nat.cast_one]
using add_val_def ϖ 1 hϖ 1 | lemma | discrete_valuation_ring.add_val_uniformizer | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"forall_true_left",
"irreducible",
"nat.cast_one",
"one_mul",
"pow_one",
"units.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_mul {a b : R} :
add_val R (a * b) = add_val R a + add_val R b | (add_val R).map_mul _ _ | lemma | discrete_valuation_ring.add_val_mul | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_pow (a : R) (n : ℕ) : add_val R (a ^ n) = n • add_val R a | (add_val R).map_pow _ _ | lemma | discrete_valuation_ring.add_val_pow | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.irreducible.add_val_pow {ϖ : R} (h : irreducible ϖ) (n : ℕ) :
add_val R (ϖ ^ n) = n | by rw [add_val_pow, add_val_uniformizer h, nsmul_one] | lemma | irreducible.add_val_pow | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"irreducible",
"nsmul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_eq_top_iff {a : R} : add_val R a = ⊤ ↔ a = 0 | begin
have hi := (classical.some_spec (exists_prime R)).irreducible,
split,
{ contrapose,
intro h,
obtain ⟨n, ha⟩ := associated_pow_irreducible h hi,
obtain ⟨u, rfl⟩ := ha.symm,
rw [mul_comm, add_val_def' u hi n],
exact part_enat.coe_ne_top _ },
{ rintro rfl,
exact add_val_zero }
end | lemma | discrete_valuation_ring.add_val_eq_top_iff | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"irreducible",
"mul_comm",
"part_enat.coe_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_le_iff_dvd {a b : R} : add_val R a ≤ add_val R b ↔ a ∣ b | begin
have hp := classical.some_spec (exists_prime R),
split; intro h,
{ by_cases ha0 : a = 0,
{ rw [ha0, add_val_zero, top_le_iff, add_val_eq_top_iff] at h,
rw h,
apply dvd_zero },
obtain ⟨n, ha⟩ := associated_pow_irreducible ha0 hp.irreducible,
rw [add_val, add_valuation_apply, add_valua... | lemma | discrete_valuation_ring.add_val_le_iff_dvd | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [
"dvd_zero",
"top_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_val_add {a b : R} :
min (add_val R a) (add_val R b) ≤ add_val R (a + b) | (add_val R).map_add _ _ | lemma | discrete_valuation_ring.add_val_add | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/basic.lean | [
"ring_theory.principal_ideal_domain",
"ring_theory.ideal.local_ring",
"ring_theory.multiplicity",
"ring_theory.valuation.basic",
"linear_algebra.adic_completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_maximal_ideal_pow_eq_of_principal [is_noetherian_ring R] [local_ring R] [is_domain R]
(h : ¬ is_field R) (h' : (maximal_ideal R).is_principal) (I : ideal R) (hI : I ≠ ⊥) :
∃ n : ℕ, I = (maximal_ideal R) ^ n | begin
classical,
unfreezingI { obtain ⟨x, hx : _ = ideal.span _⟩ := h' },
by_cases hI' : I = ⊤, { use 0, rw [pow_zero, hI', ideal.one_eq_top] },
have H : ∀ r : R, ¬ (is_unit r) ↔ x ∣ r :=
λ r, (set_like.ext_iff.mp hx r).trans ideal.mem_span_singleton,
have : x ≠ 0,
{ rintro rfl,
apply ring.ne_bot_of... | lemma | exists_maximal_ideal_pow_eq_of_principal | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/tfae.lean | [
"ring_theory.ideal.cotangent",
"ring_theory.dedekind_domain.basic",
"ring_theory.valuation.valuation_ring",
"ring_theory.nakayama"
] | [
"associated",
"associated.mul_mul",
"by_contra",
"discrete_valuation_ring.irreducible_of_span_eq_maximal_ideal",
"dvd_refl",
"eq_bot_iff",
"eq_or_ne",
"ideal",
"ideal.mem_span_singleton",
"ideal.one_eq_top",
"ideal.pow_le_pow",
"ideal.pow_mem_pow",
"ideal.span",
"ideal.span_le",
"ideal.s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maximal_ideal_is_principal_of_is_dedekind_domain
[local_ring R] [is_domain R] [is_dedekind_domain R] : (maximal_ideal R).is_principal | begin
classical,
by_cases ne_bot : maximal_ideal R = ⊥,
{ rw ne_bot, apply_instance },
obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ maximal_ideal R, a ≠ (0 : R),
{ by_contra h', push_neg at h', apply ne_bot, rwa eq_bot_iff },
have hle : ideal.span {a} ≤ maximal_ideal R,
{ rwa [ideal.span_le, set.singleton_subset_iff] },
... | lemma | maximal_ideal_is_principal_of_is_dedekind_domain | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/tfae.lean | [
"ring_theory.ideal.cotangent",
"ring_theory.dedekind_domain.basic",
"ring_theory.valuation.valuation_ring",
"ring_theory.nakayama"
] | [
"Inf_le",
"algebra.of_id",
"algebra_map",
"by_contra",
"distrib_mul_action.to_linear_map",
"div_eq_one_iff_eq",
"div_mul_cancel",
"eq_bot_iff",
"fraction_ring",
"ideal.eq_top_iff_one",
"ideal.exists_radical_pow_le_of_fg",
"ideal.is_maximal.ne_top",
"ideal.mem_span_singleton'",
"ideal.mul_m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_valuation_ring.tfae [is_noetherian_ring R] [local_ring R] [is_domain R]
(h : ¬ is_field R) :
tfae [discrete_valuation_ring R,
valuation_ring R,
is_dedekind_domain R,
is_integrally_closed R ∧ ∃! P : ideal R, P ≠ ⊥ ∧ P.is_prime,
(maximal_ideal R).is_principal,
finite_dimensional.finrank (... | begin
have ne_bot := ring.ne_bot_of_is_maximal_of_not_is_field (maximal_ideal.is_maximal R) h,
classical,
rw finrank_eq_one_iff',
tfae_have : 1 → 2,
{ introI _, apply_instance },
tfae_have : 2 → 1,
{ introI _,
haveI := is_bezout.to_gcd_domain R,
haveI : unique_factorization_monoid R := ufm_of_gcd_... | lemma | discrete_valuation_ring.tfae | ring_theory.discrete_valuation_ring | src/ring_theory/discrete_valuation_ring/tfae.lean | [
"ring_theory.ideal.cotangent",
"ring_theory.dedekind_domain.basic",
"ring_theory.valuation.valuation_ring",
"ring_theory.nakayama"
] | [
"bot_le",
"bot_ne_top",
"discrete_valuation_ring",
"discrete_valuation_ring.iff_pid_with_one_nonzero_prime",
"discrete_valuation_ring.of_ufd_of_unique_irreducible",
"exists_maximal_ideal_pow_eq_of_principal",
"finite_dimensional.finrank",
"finrank_eq_one_iff'",
"finset.coe_singleton",
"ideal",
"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_ring (𝒜 : ι → σ) extends set_like.graded_monoid 𝒜, direct_sum.decomposition 𝒜. | class | graded_ring | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"direct_sum.decomposition",
"set_like.graded_monoid"
] | An internally-graded `R`-algebra `A` is one that can be decomposed into a collection
of `submodule R A`s indexed by `ι` such that the canonical map `A → ⨁ i, 𝒜 i` is bijective and
respects multiplication, i.e. the product of an element of degree `i` and an element of degree `j`
is an element of degree `i + j`.
Note t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_ring_equiv : A ≃+* ⨁ i, 𝒜 i | ring_equiv.symm
{ map_mul' := (coe_ring_hom 𝒜).map_mul,
map_add' := (coe_ring_hom 𝒜).map_add,
..(decompose_add_equiv 𝒜).symm } | def | direct_sum.decompose_ring_equiv | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"map_mul",
"ring_equiv.symm"
] | If `A` is graded by `ι` with degree `i` component `𝒜 i`, then it is isomorphic as
a ring to a direct sum of components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decompose_one : decompose 𝒜 (1 : A) = 1 | map_one (decompose_ring_equiv 𝒜) | lemma | direct_sum.decompose_one | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_symm_one : (decompose 𝒜).symm 1 = (1 : A) | map_one (decompose_ring_equiv 𝒜).symm | lemma | direct_sum.decompose_symm_one | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_mul (x y : A) : decompose 𝒜 (x * y) = decompose 𝒜 x * decompose 𝒜 y | map_mul (decompose_ring_equiv 𝒜) x y | lemma | direct_sum.decompose_mul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_symm_mul (x y : ⨁ i, 𝒜 i) :
(decompose 𝒜).symm (x * y) = (decompose 𝒜).symm x * (decompose 𝒜).symm y | map_mul (decompose_ring_equiv 𝒜).symm x y | lemma | direct_sum.decompose_symm_mul | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_ring.proj (i : ι) : A →+ A | (add_submonoid_class.subtype (𝒜 i)).comp $
(dfinsupp.eval_add_monoid_hom i).comp $
ring_hom.to_add_monoid_hom $ ring_equiv.to_ring_hom $ direct_sum.decompose_ring_equiv 𝒜 | def | graded_ring.proj | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"dfinsupp.eval_add_monoid_hom",
"direct_sum.decompose_ring_equiv",
"ring_equiv.to_ring_hom"
] | The projection maps of a graded ring | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
graded_ring.proj_apply (i : ι) (r : A) :
graded_ring.proj 𝒜 i r = (decompose 𝒜 r : ⨁ i, 𝒜 i) i | rfl | lemma | graded_ring.proj_apply | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"graded_ring.proj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_ring.proj_recompose (a : ⨁ i, 𝒜 i) (i : ι) :
graded_ring.proj 𝒜 i ((decompose 𝒜).symm a) =
(decompose 𝒜).symm (direct_sum.of _ i (a i)) | by rw [graded_ring.proj_apply, decompose_symm_of, equiv.apply_symm_apply] | lemma | graded_ring.proj_recompose | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"direct_sum.of",
"equiv.apply_symm_apply",
"graded_ring.proj",
"graded_ring.proj_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_ring.mem_support_iff [Π i (x : 𝒜 i), decidable (x ≠ 0)] (r : A) (i : ι) :
i ∈ (decompose 𝒜 r).support ↔ graded_ring.proj 𝒜 i r ≠ 0 | dfinsupp.mem_support_iff.trans zero_mem_class.coe_eq_zero.not.symm | lemma | graded_ring.mem_support_iff | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"graded_ring.proj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_decompose_mul_add_of_left_mem
[add_left_cancel_monoid ι] [graded_ring 𝒜] {a b : A} (a_mem : a ∈ 𝒜 i) :
(decompose 𝒜 (a * b) (i + j) : A) = a * decompose 𝒜 b j | by { lift a to 𝒜 i using a_mem, rw [decompose_mul, decompose_coe, coe_of_mul_apply_add] } | lemma | direct_sum.coe_decompose_mul_add_of_left_mem | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"add_left_cancel_monoid",
"graded_ring",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_decompose_mul_add_of_right_mem
[add_right_cancel_monoid ι] [graded_ring 𝒜] {a b : A} (b_mem : b ∈ 𝒜 j) :
(decompose 𝒜 (a * b) (i + j) : A) = decompose 𝒜 a i * b | by { lift b to 𝒜 j using b_mem, rw [decompose_mul, decompose_coe, coe_mul_of_apply_add] } | lemma | direct_sum.coe_decompose_mul_add_of_right_mem | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"add_right_cancel_monoid",
"graded_ring",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_mul_add_left
[add_left_cancel_monoid ι] [graded_ring 𝒜] (a : 𝒜 i) {b : A} :
decompose 𝒜 (↑a * b) (i + j) =
@graded_monoid.ghas_mul.mul ι (λ i, 𝒜 i) _ _ _ _ a (decompose 𝒜 b j) | subtype.ext $ coe_decompose_mul_add_of_left_mem 𝒜 a.2 | lemma | direct_sum.decompose_mul_add_left | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"add_left_cancel_monoid",
"graded_ring",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decompose_mul_add_right
[add_right_cancel_monoid ι] [graded_ring 𝒜] {a : A} (b : 𝒜 j) :
decompose 𝒜 (a * ↑b) (i + j) =
@graded_monoid.ghas_mul.mul ι (λ i, 𝒜 i) _ _ _ _ (decompose 𝒜 a i) b | subtype.ext $ coe_decompose_mul_add_of_right_mem 𝒜 b.2 | lemma | direct_sum.decompose_mul_add_right | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"add_right_cancel_monoid",
"graded_ring",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_algebra | graded_ring 𝒜 | def | graded_algebra | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"graded_ring"
] | A special case of `graded_ring` with `σ = submodule R A`. This is useful both because it
can avoid typeclass search, and because it provides a more concise name. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
graded_algebra.of_alg_hom [set_like.graded_monoid 𝒜] (decompose : A →ₐ[R] ⨁ i, 𝒜 i)
(right_inv : (direct_sum.coe_alg_hom 𝒜).comp decompose = alg_hom.id R A)
(left_inv : ∀ i (x : 𝒜 i), decompose (x : A) = direct_sum.of (λ i, ↥(𝒜 i)) i x) :
graded_algebra 𝒜 | { decompose' := decompose,
left_inv := alg_hom.congr_fun right_inv,
right_inv := begin
suffices : decompose.comp (direct_sum.coe_alg_hom 𝒜) = alg_hom.id _ _,
from alg_hom.congr_fun this,
ext i x : 2,
exact (decompose.congr_arg $ direct_sum.coe_alg_hom_of _ _ _).trans (left_inv i x),
end} | def | graded_algebra.of_alg_hom | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"alg_hom.congr_fun",
"alg_hom.id",
"direct_sum.coe_alg_hom",
"direct_sum.coe_alg_hom_of",
"direct_sum.of",
"graded_algebra",
"set_like.graded_monoid"
] | A helper to construct a `graded_algebra` when the `set_like.graded_monoid` structure is already
available. This makes the `left_inv` condition easier to prove, and phrases the `right_inv`
condition in a way that allows custom `@[ext]` lemmas to apply.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decompose_alg_equiv : A ≃ₐ[R] ⨁ i, 𝒜 i | alg_equiv.symm
{ map_mul' := (coe_alg_hom 𝒜).map_mul,
map_add' := (coe_alg_hom 𝒜).map_add,
commutes' := (coe_alg_hom 𝒜).commutes,
..(decompose_add_equiv 𝒜).symm } | def | direct_sum.decompose_alg_equiv | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"alg_equiv.symm",
"map_mul"
] | If `A` is graded by `ι` with degree `i` component `𝒜 i`, then it is isomorphic as
an algebra to a direct sum of components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
graded_algebra.proj (𝒜 : ι → submodule R A) [graded_algebra 𝒜] (i : ι) : A →ₗ[R] A | (𝒜 i).subtype.comp $
(dfinsupp.lapply i).comp $
(decompose_alg_equiv 𝒜).to_alg_hom.to_linear_map | def | graded_algebra.proj | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"dfinsupp.lapply",
"graded_algebra",
"submodule"
] | The projection maps of graded algebra | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
graded_algebra.proj_apply (i : ι) (r : A) :
graded_algebra.proj 𝒜 i r = (decompose 𝒜 r : ⨁ i, 𝒜 i) i | rfl | lemma | graded_algebra.proj_apply | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"graded_algebra.proj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_algebra.proj_recompose (a : ⨁ i, 𝒜 i) (i : ι) :
graded_algebra.proj 𝒜 i ((decompose 𝒜).symm a) =
(decompose 𝒜).symm (of _ i (a i)) | by rw [graded_algebra.proj_apply, decompose_symm_of, equiv.apply_symm_apply] | lemma | graded_algebra.proj_recompose | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"equiv.apply_symm_apply",
"graded_algebra.proj",
"graded_algebra.proj_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_algebra.mem_support_iff [decidable_eq A] (r : A) (i : ι) :
i ∈ (decompose 𝒜 r).support ↔ graded_algebra.proj 𝒜 i r ≠ 0 | dfinsupp.mem_support_iff.trans submodule.coe_eq_zero.not.symm | lemma | graded_algebra.mem_support_iff | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"graded_algebra.proj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
graded_ring.proj_zero_ring_hom : A →+* A | { to_fun := λ a, decompose 𝒜 a 0,
map_one' := decompose_of_mem_same 𝒜 one_mem,
map_zero' := by { rw decompose_zero, refl },
map_add' := λ _ _, by { rw decompose_add, refl },
map_mul' := begin
refine direct_sum.decomposition.induction_on 𝒜 (λ x, _) _ _,
{ simp only [zero_mul, decompose_zero, zero_appl... | def | graded_ring.proj_zero_ring_hom | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"direct_sum.decomposition.induction_on",
"mul_zero",
"not_and_distrib",
"subtype.coe_mk",
"zero_mul"
] | If `A` is graded by a canonically ordered add monoid, then the projection map `x ↦ x₀` is a ring
homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_decompose_mul_of_left_mem_of_not_le
(a_mem : a ∈ 𝒜 i) (h : ¬ i ≤ n) : (decompose 𝒜 (a * b) n : A) = 0 | by { lift a to 𝒜 i using a_mem, rwa [decompose_mul, decompose_coe, coe_of_mul_apply_of_not_le] } | lemma | direct_sum.coe_decompose_mul_of_left_mem_of_not_le | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_decompose_mul_of_right_mem_of_not_le
(b_mem : b ∈ 𝒜 i) (h : ¬ i ≤ n) : (decompose 𝒜 (a * b) n : A) = 0 | by { lift b to 𝒜 i using b_mem, rwa [decompose_mul, decompose_coe, coe_mul_of_apply_of_not_le] } | lemma | direct_sum.coe_decompose_mul_of_right_mem_of_not_le | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_decompose_mul_of_left_mem_of_le
(a_mem : a ∈ 𝒜 i) (h : i ≤ n) : (decompose 𝒜 (a * b) n : A) = a * decompose 𝒜 b (n - i) | by { lift a to 𝒜 i using a_mem, rwa [decompose_mul, decompose_coe, coe_of_mul_apply_of_le] } | lemma | direct_sum.coe_decompose_mul_of_left_mem_of_le | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_decompose_mul_of_right_mem_of_le
(b_mem : b ∈ 𝒜 i) (h : i ≤ n) : (decompose 𝒜 (a * b) n : A) = decompose 𝒜 a (n - i) * b | by { lift b to 𝒜 i using b_mem, rwa [decompose_mul, decompose_coe, coe_mul_of_apply_of_le] } | lemma | direct_sum.coe_decompose_mul_of_right_mem_of_le | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_decompose_mul_of_left_mem (n) [decidable (i ≤ n)] (a_mem : a ∈ 𝒜 i) :
(decompose 𝒜 (a * b) n : A) = if i ≤ n then a * decompose 𝒜 b (n - i) else 0 | by { lift a to 𝒜 i using a_mem, rwa [decompose_mul, decompose_coe, coe_of_mul_apply] } | lemma | direct_sum.coe_decompose_mul_of_left_mem | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_decompose_mul_of_right_mem (n) [decidable (i ≤ n)] (b_mem : b ∈ 𝒜 i) :
(decompose 𝒜 (a * b) n : A) = if i ≤ n then decompose 𝒜 a (n - i) * b else 0 | by { lift b to 𝒜 i using b_mem, rwa [decompose_mul, decompose_coe, coe_mul_of_apply] } | lemma | direct_sum.coe_decompose_mul_of_right_mem | ring_theory.graded_algebra | src/ring_theory/graded_algebra/basic.lean | [
"algebra.direct_sum.algebra",
"algebra.direct_sum.decomposition",
"algebra.direct_sum.internal",
"algebra.direct_sum.ring"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous : Prop | ∀ (i : ι) ⦃r : A⦄, r ∈ I → (direct_sum.decompose 𝒜 r i : A) ∈ I | def | ideal.is_homogeneous | 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"
] | An `I : ideal A` is homogeneous if for every `r ∈ I`, all homogeneous components
of `r` are in `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_ideal extends submodule A A | (is_homogeneous' : ideal.is_homogeneous 𝒜 to_submodule) | structure | homogeneous_ideal | 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.is_homogeneous",
"submodule"
] | For any `semiring A`, we collect the homogeneous ideals of `A` into a type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_ideal.to_ideal (I : homogeneous_ideal 𝒜) : ideal A | I.to_submodule | def | homogeneous_ideal.to_ideal | 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"
] | Converting a homogeneous ideal to an ideal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homogeneous_ideal.is_homogeneous (I : homogeneous_ideal 𝒜) :
I.to_ideal.is_homogeneous 𝒜 | I.is_homogeneous' | lemma | homogeneous_ideal.is_homogeneous | 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 | |
homogeneous_ideal.to_ideal_injective :
function.injective (homogeneous_ideal.to_ideal : homogeneous_ideal 𝒜 → ideal A) | λ ⟨x, hx⟩ ⟨y, hy⟩ (h : x = y), by simp [h] | lemma | homogeneous_ideal.to_ideal_injective | 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",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_ideal.set_like : set_like (homogeneous_ideal 𝒜) A | { coe := λ I, I.to_ideal,
coe_injective' := λ I J h, homogeneous_ideal.to_ideal_injective $ set_like.coe_injective h } | instance | homogeneous_ideal.set_like | 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",
"set_like",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_ideal.ext {I J : homogeneous_ideal 𝒜}
(h : I.to_ideal = J.to_ideal) : I = J | homogeneous_ideal.to_ideal_injective h | lemma | homogeneous_ideal.ext | 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 | |
homogeneous_ideal.mem_iff {I : homogeneous_ideal 𝒜} {x : A} :
x ∈ I.to_ideal ↔ x ∈ I | iff.rfl | lemma | homogeneous_ideal.mem_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_core' (I : ideal A) : ideal A | ideal.span (coe '' ((coe : subtype (is_homogeneous 𝒜) → A) ⁻¹' I)) | def | ideal.homogeneous_core' | 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.span"
] | For any `I : ideal A`, not necessarily homogeneous, `I.homogeneous_core' 𝒜`
is the largest homogeneous ideal of `A` contained in `I`, as an ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.homogeneous_core'_mono : monotone (ideal.homogeneous_core' 𝒜) | λ I J I_le_J, ideal.span_mono $ set.image_subset _ $ λ x, @I_le_J _ | lemma | ideal.homogeneous_core'_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_core'",
"ideal.span_mono",
"monotone",
"set.image_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_core'_le : I.homogeneous_core' 𝒜 ≤ I | ideal.span_le.2 $ image_preimage_subset _ _ | lemma | ideal.homogeneous_core'_le | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous_iff_forall_subset :
I.is_homogeneous 𝒜 ↔ ∀ i, (I : set A) ⊆ graded_ring.proj 𝒜 i ⁻¹' I | iff.rfl | lemma | ideal.is_homogeneous_iff_forall_subset | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous_iff_subset_Inter :
I.is_homogeneous 𝒜 ↔ (I : set A) ⊆ ⋂ i, graded_ring.proj 𝒜 i ⁻¹' ↑I | subset_Inter_iff.symm | lemma | ideal.is_homogeneous_iff_subset_Inter | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.mul_homogeneous_element_mem_of_mem
{I : ideal A} (r x : A) (hx₁ : is_homogeneous 𝒜 x) (hx₂ : x ∈ I) (j : ι) :
graded_ring.proj 𝒜 j (r * x) ∈ I | begin
classical,
rw [←direct_sum.sum_support_decompose 𝒜 r, finset.sum_mul, map_sum],
apply ideal.sum_mem,
intros k hk,
obtain ⟨i, hi⟩ := hx₁,
have mem₁ : (direct_sum.decompose 𝒜 r k : A) * x ∈ 𝒜 (k + i) := graded_monoid.mul_mem
(set_like.coe_mem _) hi,
erw [graded_ring.proj_apply, direct_sum.decom... | lemma | ideal.mul_homogeneous_element_mem_of_mem | 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",
"direct_sum.decompose_of_mem",
"finset.sum_mul",
"graded_ring.proj",
"graded_ring.proj_apply",
"ideal",
"ideal.sum_mem",
"set_like.coe_mem",
"set_like.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous_span (s : set A) (h : ∀ x ∈ s, is_homogeneous 𝒜 x) :
(ideal.span s).is_homogeneous 𝒜 | begin
rintros i r hr,
rw [ideal.span, finsupp.span_eq_range_total] at hr,
rw linear_map.mem_range at hr,
obtain ⟨s, rfl⟩ := hr,
rw [finsupp.total_apply, finsupp.sum, decompose_sum, dfinsupp.finset_sum_apply,
add_submonoid_class.coe_finset_sum],
refine ideal.sum_mem _ _,
rintros z hz1,
rw [smul_eq_mu... | lemma | ideal.is_homogeneous_span | 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"
] | [
"dfinsupp.finset_sum_apply",
"finsupp.span_eq_range_total",
"finsupp.total_apply",
"ideal.mul_homogeneous_element_mem_of_mem",
"ideal.span",
"ideal.subset_span",
"ideal.sum_mem",
"linear_map.mem_range",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.homogeneous_core : homogeneous_ideal 𝒜 | ⟨ideal.homogeneous_core' 𝒜 I,
ideal.is_homogeneous_span _ _ (λ x h, by { rw [subtype.image_preimage_coe] at h, exact h.2 })⟩ | def | ideal.homogeneous_core | 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.is_homogeneous_span",
"subtype.image_preimage_coe"
] | For any `I : ideal A`, not necessarily homogeneous, `I.homogeneous_core' 𝒜`
is the largest homogeneous ideal of `A` contained in `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.homogeneous_core_mono : monotone (ideal.homogeneous_core 𝒜) | ideal.homogeneous_core'_mono 𝒜 | lemma | ideal.homogeneous_core_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_core",
"ideal.homogeneous_core'_mono",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.to_ideal_homogeneous_core_le : (I.homogeneous_core 𝒜).to_ideal ≤ I | ideal.homogeneous_core'_le 𝒜 I | lemma | ideal.to_ideal_homogeneous_core_le | 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_core'_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.mem_homogeneous_core_of_is_homogeneous_of_mem {x : A}
(h : set_like.is_homogeneous 𝒜 x) (hmem : x ∈ I) : x ∈ I.homogeneous_core 𝒜 | ideal.subset_span ⟨⟨x, h⟩, hmem, rfl⟩ | lemma | ideal.mem_homogeneous_core_of_is_homogeneous_of_mem | 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.subset_span",
"set_like.is_homogeneous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous.to_ideal_homogeneous_core_eq_self (h : I.is_homogeneous 𝒜) :
(I.homogeneous_core 𝒜).to_ideal = I | begin
apply le_antisymm (I.homogeneous_core'_le 𝒜) _,
intros x hx,
classical,
rw ←direct_sum.sum_support_decompose 𝒜 x,
exact ideal.sum_mem _ (λ j hj, ideal.subset_span ⟨⟨_, is_homogeneous_coe _⟩, h _ hx, rfl⟩)
end | lemma | ideal.is_homogeneous.to_ideal_homogeneous_core_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.subset_span",
"ideal.sum_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homogeneous_ideal.to_ideal_homogeneous_core_eq_self (I : homogeneous_ideal 𝒜) :
I.to_ideal.homogeneous_core 𝒜 = I | by ext1; convert ideal.is_homogeneous.to_ideal_homogeneous_core_eq_self I.is_homogeneous | lemma | homogeneous_ideal.to_ideal_homogeneous_core_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",
"ideal.is_homogeneous.to_ideal_homogeneous_core_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous.iff_eq : I.is_homogeneous 𝒜 ↔ (I.homogeneous_core 𝒜).to_ideal = I | ⟨ λ hI, hI.to_ideal_homogeneous_core_eq_self,
λ hI, hI ▸ (ideal.homogeneous_core 𝒜 I).2 ⟩ | lemma | ideal.is_homogeneous.iff_eq | 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_core"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_homogeneous.iff_exists :
I.is_homogeneous 𝒜 ↔ ∃ (S : set (homogeneous_submonoid 𝒜)), I = ideal.span (coe '' S) | begin
rw [ideal.is_homogeneous.iff_eq, eq_comm],
exact ((set.image_preimage.compose (submodule.gi _ _).gc).exists_eq_l _).symm,
end | lemma | ideal.is_homogeneous.iff_exists | 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.is_homogeneous.iff_eq",
"ideal.span",
"submodule.gi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot : ideal.is_homogeneous 𝒜 ⊥ | λ i r hr,
begin
simp only [ideal.mem_bot] at hr,
rw [hr, decompose_zero, zero_apply],
apply ideal.zero_mem
end | lemma | ideal.is_homogeneous.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"
] | [
"ideal.is_homogeneous",
"ideal.mem_bot",
"ideal.zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top : ideal.is_homogeneous 𝒜 ⊤ | λ i r hr, by simp only [submodule.mem_top] | lemma | ideal.is_homogeneous.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"
] | [
"ideal.is_homogeneous",
"submodule.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf {I J : ideal A} (HI : I.is_homogeneous 𝒜) (HJ : J.is_homogeneous 𝒜) :
(I ⊓ J).is_homogeneous 𝒜 | λ i r hr, ⟨HI _ hr.1, HJ _ hr.2⟩ | lemma | ideal.is_homogeneous.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"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup {I J : ideal A} (HI : I.is_homogeneous 𝒜) (HJ : J.is_homogeneous 𝒜) :
(I ⊔ J).is_homogeneous 𝒜 | begin
rw iff_exists at HI HJ ⊢,
obtain ⟨⟨s₁, rfl⟩, ⟨s₂, rfl⟩⟩ := ⟨HI, HJ⟩,
refine ⟨s₁ ∪ s₂, _⟩,
rw [set.image_union],
exact (submodule.span_union _ _).symm,
end | lemma | ideal.is_homogeneous.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"
] | [
"ideal",
"set.image_union",
"submodule.span_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr {κ : Sort*} {f : κ → ideal A} (h : ∀ i, (f i).is_homogeneous 𝒜) :
(⨆ i, f i).is_homogeneous 𝒜 | begin
simp_rw iff_exists at h ⊢,
choose s hs using h,
refine ⟨⋃ i, s i, _⟩,
simp_rw [set.image_Union, ideal.span_Union],
congr',
exact funext hs,
end | lemma | ideal.is_homogeneous.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"
] | [
"ideal",
"ideal.span_Union",
"set.image_Union",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi {κ : Sort*} {f : κ → ideal A} (h : ∀ i, (f i).is_homogeneous 𝒜) :
(⨅ i, f i).is_homogeneous 𝒜 | begin
intros i x hx,
simp only [ideal.mem_infi] at ⊢ hx,
exact λ j, h _ _ (hx j),
end | lemma | ideal.is_homogeneous.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"
] | [
"ideal",
"ideal.mem_infi",
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr₂ {κ : Sort*} {κ' : κ → Sort*} {f : Π i, κ' i → ideal A}
(h : ∀ i j, (f i j).is_homogeneous 𝒜) :
(⨆ i j, f i j).is_homogeneous 𝒜 | is_homogeneous.supr $ λ i, is_homogeneous.supr $ h i | lemma | ideal.is_homogeneous.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"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi₂ {κ : Sort*} {κ' : κ → Sort*} {f : Π i, κ' i → ideal A}
(h : ∀ i j, (f i j).is_homogeneous 𝒜) :
(⨅ i j, f i j).is_homogeneous 𝒜 | is_homogeneous.infi $ λ i, is_homogeneous.infi $ h i | lemma | ideal.is_homogeneous.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"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup {ℐ : set (ideal A)} (h : ∀ I ∈ ℐ, ideal.is_homogeneous 𝒜 I) :
(Sup ℐ).is_homogeneous 𝒜 | by { rw Sup_eq_supr, exact supr₂ h } | lemma | ideal.is_homogeneous.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"
] | [
"Sup_eq_supr",
"ideal",
"ideal.is_homogeneous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf {ℐ : set (ideal A)} (h : ∀ I ∈ ℐ, ideal.is_homogeneous 𝒜 I) :
(Inf ℐ).is_homogeneous 𝒜 | by { rw Inf_eq_infi, exact infi₂ h } | lemma | ideal.is_homogeneous.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"
] | [
"Inf_eq_infi",
"ideal",
"ideal.is_homogeneous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ((⊤ : homogeneous_ideal 𝒜) : set A) = univ | rfl | lemma | homogeneous_ideal.coe_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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