statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
integral_closure_le_span_dual_basis [is_separable K L]
{ι : Type*} [fintype ι] [decidable_eq ι] (b : basis ι K L)
(hb_int : ∀ i, is_integral A (b i)) [is_integrally_closed A] :
(integral_closure A L).to_submodule ≤ submodule.span A (set.range $
(trace_form K L).dual_basis (trace_form_nondegenerate K L) b) | begin
refine le_trans _ (is_integral_closure.range_le_span_dual_basis (integral_closure A L) b hb_int),
intros x hx,
exact ⟨⟨x, hx⟩, rfl⟩
end | lemma | integral_closure_le_span_dual_basis | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"basis",
"fintype",
"integral_closure",
"is_integral",
"is_integral_closure.range_le_span_dual_basis",
"is_integrally_closed",
"is_separable",
"set.range",
"submodule.span",
"trace_form_nondegenerate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_integral_multiples (s : finset L) :
∃ (y ≠ (0 : A)), ∀ x ∈ s, is_integral A (y • x) | begin
haveI := classical.dec_eq L,
refine s.induction _ _,
{ use [1, one_ne_zero],
rintros x ⟨⟩ },
{ rintros x s hx ⟨y, hy, hs⟩,
obtain ⟨x', y', hy', hx'⟩ := exists_integral_multiple
((is_fraction_ring.is_algebraic_iff A K L).mpr (is_algebraic_of_finite _ _ x))
((injective_iff_map_eq_zero (a... | lemma | exists_integral_multiples | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"algebra.smul_def",
"algebra_map",
"classical.dec_eq",
"exists_integral_multiple",
"finset",
"is_fraction_ring.injective",
"is_fraction_ring.is_algebraic_iff",
"is_integral",
"is_integral_algebra_map",
"is_integral_mul",
"is_scalar_tower.algebra_map_eq",
"mul_comm",
"mul_ne_zero",
"one_ne_... | Send a set of `x`'es in a finite extension `L` of the fraction field of `R`
to `(y : R) • x ∈ integral_closure R L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_dimensional.exists_is_basis_integral :
∃ (s : finset L) (b : basis s K L), (∀ x, is_integral A (b x)) | begin
letI := classical.dec_eq L,
letI : is_noetherian K L := is_noetherian.iff_fg.2 infer_instance,
let s' := is_noetherian.finset_basis_index K L,
let bs' := is_noetherian.finset_basis K L,
obtain ⟨y, hy, his'⟩ := exists_integral_multiples A K (finset.univ.image bs'),
have hy' : algebra_map A L y ≠ 0,
{... | lemma | finite_dimensional.exists_is_basis_integral | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"algebra.lmul",
"algebra.smul_def",
"algebra_map",
"basis",
"basis.map_apply",
"classical.dec_eq",
"exists_integral_multiples",
"exists_prop",
"finset",
"finset.mem_image",
"finset.mem_univ",
"inv_fun",
"inv_mul_cancel_left₀",
"is_fraction_ring.injective",
"is_integral",
"is_noetherian... | If `L` is a finite extension of `K = Frac(A)`,
then `L` has a basis over `A` consisting of integral elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_closure.is_noetherian [is_integrally_closed A] [is_noetherian_ring A] :
is_noetherian A C | begin
haveI := classical.dec_eq L,
obtain ⟨s, b, hb_int⟩ := finite_dimensional.exists_is_basis_integral A K L,
let b' := (trace_form K L).dual_basis (trace_form_nondegenerate K L) b,
letI := is_noetherian_span_of_finite A (set.finite_range b'),
let f : C →ₗ[A] submodule.span A (set.range b') :=
(submodule... | lemma | is_integral_closure.is_noetherian | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"algebra.linear_map",
"classical.dec_eq",
"finite_dimensional.exists_is_basis_integral",
"is_integral_closure.range_le_span_dual_basis",
"is_integrally_closed",
"is_noetherian",
"is_noetherian_of_ker_bot",
"is_noetherian_ring",
"is_noetherian_span_of_finite",
"linear_map.ker_cod_restrict",
"line... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.is_noetherian_ring [is_integrally_closed A] [is_noetherian_ring A] :
is_noetherian_ring C | is_noetherian_ring_iff.mpr $ is_noetherian_of_tower A (is_integral_closure.is_noetherian A K L C) | lemma | is_integral_closure.is_noetherian_ring | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"is_integral_closure.is_noetherian",
"is_integrally_closed",
"is_noetherian_of_tower",
"is_noetherian_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.module_free [no_zero_smul_divisors A L] [is_principal_ideal_ring A] :
module.free A C | begin
haveI : no_zero_smul_divisors A C := is_integral_closure.no_zero_smul_divisors A L,
haveI : is_noetherian A C := is_integral_closure.is_noetherian A K L _,
exact module.free_of_finite_type_torsion_free',
end | lemma | is_integral_closure.module_free | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"is_integral_closure.is_noetherian",
"is_integral_closure.no_zero_smul_divisors",
"is_noetherian",
"is_principal_ideal_ring",
"module.free",
"module.free_of_finite_type_torsion_free'",
"no_zero_smul_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.rank [is_principal_ideal_ring A] [no_zero_smul_divisors A L] :
finite_dimensional.finrank A C = finite_dimensional.finrank K L | begin
haveI : module.free A C := is_integral_closure.module_free A K L C,
haveI : is_noetherian A C := is_integral_closure.is_noetherian A K L C,
haveI : is_localization (algebra.algebra_map_submonoid C A⁰) L :=
is_integral_closure.is_localization A K L C,
let b := basis.localization_localization K A⁰ L (mo... | lemma | is_integral_closure.rank | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"algebra.algebra_map_submonoid",
"basis.localization_localization",
"finite_dimensional.finrank",
"finite_dimensional.finrank_eq_card_basis",
"finite_dimensional.finrank_eq_card_choose_basis_index",
"is_integral_closure.is_localization",
"is_integral_closure.is_noetherian",
"is_integral_closure.module... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure.is_noetherian_ring [is_integrally_closed A] [is_noetherian_ring A] :
is_noetherian_ring (integral_closure A L) | is_integral_closure.is_noetherian_ring A K L (integral_closure A L) | lemma | integral_closure.is_noetherian_ring | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"integral_closure",
"is_integral_closure.is_noetherian_ring",
"is_integrally_closed",
"is_noetherian_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.is_dedekind_domain [h : is_dedekind_domain A] :
is_dedekind_domain C | begin
haveI : is_fraction_ring C L := is_integral_closure.is_fraction_ring_of_finite_extension A K L C,
exact
⟨is_integral_closure.is_noetherian_ring A K L C,
h.dimension_le_one.is_integral_closure _ L _,
(is_integrally_closed_iff L).mpr (λ x hx, ⟨is_integral_closure.mk' C x
(is_integral_trans (is_int... | lemma | is_integral_closure.is_dedekind_domain | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"is_dedekind_domain",
"is_fraction_ring",
"is_integral_closure.algebra_map_mk'",
"is_integral_closure.is_fraction_ring_of_finite_extension",
"is_integral_closure.is_integral_algebra",
"is_integral_trans",
"is_integrally_closed_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure.is_dedekind_domain [h : is_dedekind_domain A] :
is_dedekind_domain (integral_closure A L) | is_integral_closure.is_dedekind_domain A K L (integral_closure A L) | lemma | integral_closure.is_dedekind_domain | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"integral_closure",
"is_dedekind_domain",
"is_integral_closure.is_dedekind_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure.is_dedekind_domain_fraction_ring
[is_dedekind_domain A] : is_dedekind_domain (integral_closure A L) | integral_closure.is_dedekind_domain A (fraction_ring A) L | instance | integral_closure.is_dedekind_domain_fraction_ring | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/integral_closure.lean | [
"linear_algebra.free_module.pid",
"ring_theory.dedekind_domain.basic",
"ring_theory.localization.module",
"ring_theory.trace"
] | [
"fraction_ring",
"integral_closure",
"integral_closure.is_dedekind_domain",
"is_dedekind_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne
{P : ideal R} (hP : P.is_prime) [is_domain R] [is_dedekind_domain R]
{x : R} (x_mem : x ∈ P) (hxP2 : x ∉ P^2)
(hxQ : ∀ (Q : ideal R), is_prime Q → Q ≠ P → x ∉ Q) :
P = ideal.span {x} | begin
letI := classical.dec_eq (ideal R),
have hx0 : x ≠ 0,
{ rintro rfl,
exact hxP2 (zero_mem _) },
by_cases hP0 : P = ⊥,
{ unfreezingI { subst hP0 },
simpa using hxP2 },
have hspan0 : span ({x} : set R) ≠ ⊥ := mt ideal.span_singleton_eq_bot.mp hx0,
have span_le := (ideal.span_singleton_le_iff_me... | lemma | ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/pid.lean | [
"ring_theory.dedekind_domain.dvr",
"ring_theory.dedekind_domain.ideal"
] | [
"classical.dec_eq",
"ideal",
"ideal.count_normalized_factors_eq",
"ideal.dvd_iff_le",
"ideal.prime_of_is_prime",
"ideal.span",
"ideal.span_singleton_le_iff_mem",
"irreducible",
"is_dedekind_domain",
"is_domain",
"multiset.count_singleton",
"multiset.le_iff_count",
"normalize_eq",
"pow_one"... | Let `P` be a prime ideal, `x ∈ P \ P²` and `x ∉ Q` for all prime ideals `Q ≠ P`.
Then `P` is generated by `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fractional_ideal.is_principal_of_unit_of_comap_mul_span_singleton_eq_top
{R A : Type*} [comm_ring R] [comm_ring A] [algebra R A] {S : submonoid R} [is_localization S A]
(I : (fractional_ideal S A)ˣ) {v : A} (hv : v ∈ (↑I⁻¹ : fractional_ideal S A))
(h : submodule.comap (algebra.linear_map R A) (I * submodule.span ... | begin
have hinv := I.mul_inv,
set J := submodule.comap (algebra.linear_map R A) (I * submodule.span R {v}),
have hJ : is_localization.coe_submodule A J = I * submodule.span R {v},
{ rw [subtype.ext_iff, fractional_ideal.coe_mul, fractional_ideal.coe_one] at hinv,
apply submodule.map_comap_eq_self,
rw [←... | lemma | fractional_ideal.is_principal_of_unit_of_comap_mul_span_singleton_eq_top | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/pid.lean | [
"ring_theory.dedekind_domain.dvr",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra",
"algebra.linear_map",
"algebra_map",
"coe_coe",
"comm_ring",
"fractional_ideal",
"fractional_ideal.coe_mul",
"fractional_ideal.coe_one",
"fractional_ideal.coe_span_singleton",
"fractional_ideal.mem_span_singleton_self",
"fractional_ideal.mul_le_mul_left",
"fractional_ideal.mul_mem_m... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fractional_ideal.is_principal.of_finite_maximals_of_inv
{A : Type*} [comm_ring A] [algebra R A] {S : submonoid R} [is_localization S A]
(hS : S ≤ R⁰) (hf : {I : ideal R | I.is_maximal}.finite)
(I I' : fractional_ideal S A) (hinv : I * I' = 1) :
submodule.is_principal (I : submodule R A) | begin
have hinv' := hinv,
rw [subtype.ext_iff, fractional_ideal.coe_mul] at hinv,
let s := hf.to_finset,
haveI := classical.dec_eq (ideal R),
have coprime : ∀ (M ∈ s) (M' ∈ s.erase M), M ⊔ M' = ⊤,
{ simp_rw [finset.mem_erase, hf.mem_to_finset],
rintro M hM M' ⟨hne, hM'⟩,
exact ideal.is_maximal.copri... | theorem | fractional_ideal.is_principal.of_finite_maximals_of_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/pid.lean | [
"ring_theory.dedekind_domain.dvr",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"classical.dec_eq",
"comm_ring",
"finite",
"finset.mem_erase",
"finset.mul_sum",
"fractional_ideal",
"fractional_ideal.coe_mul",
"fractional_ideal.is_principal_of_unit_of_comap_mul_span_singleton_eq_top",
"ideal",
"ideal.exists_le_maximal",
"id... | An invertible fractional ideal of a commutative ring with finitely many maximal ideals is principal.
https://math.stackexchange.com/a/95857 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.is_principal.of_finite_maximals_of_is_unit
(hf : {I : ideal R | I.is_maximal}.finite)
{I : ideal R} (hI : is_unit (I : fractional_ideal R⁰ (fraction_ring R))) :
I.is_principal | (is_localization.coe_submodule_is_principal _ le_rfl).mp
(fractional_ideal.is_principal.of_finite_maximals_of_inv le_rfl hf I
(↑(hI.unit⁻¹) : fractional_ideal R⁰ (fraction_ring R))
hI.unit.mul_inv) | theorem | ideal.is_principal.of_finite_maximals_of_is_unit | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/pid.lean | [
"ring_theory.dedekind_domain.dvr",
"ring_theory.dedekind_domain.ideal"
] | [
"finite",
"fraction_ring",
"fractional_ideal",
"fractional_ideal.is_principal.of_finite_maximals_of_inv",
"ideal",
"is_localization.coe_submodule_is_principal",
"is_unit",
"le_rfl"
] | An invertible ideal in a commutative ring with finitely many maximal ideals is principal.
https://math.stackexchange.com/a/95857 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_principal_ideal_ring.of_finite_primes [is_domain R] [is_dedekind_domain R]
(h : {I : ideal R | I.is_prime}.finite) :
is_principal_ideal_ring R | ⟨λ I, begin
obtain rfl | hI := eq_or_ne I ⊥,
{ exact bot_is_principal },
apply ideal.is_principal.of_finite_maximals_of_is_unit,
{ apply h.subset, exact @ideal.is_maximal.is_prime _ _ },
{ exact is_unit_of_mul_eq_one _ _ (fractional_ideal.coe_ideal_mul_inv I hI) },
end⟩ | theorem | is_principal_ideal_ring.of_finite_primes | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/pid.lean | [
"ring_theory.dedekind_domain.dvr",
"ring_theory.dedekind_domain.ideal"
] | [
"bot_is_principal",
"eq_or_ne",
"finite",
"fractional_ideal.coe_ideal_mul_inv",
"ideal",
"ideal.is_maximal.is_prime",
"ideal.is_principal.of_finite_maximals_of_is_unit",
"is_dedekind_domain",
"is_domain",
"is_principal_ideal_ring",
"is_unit_of_mul_eq_one"
] | A Dedekind domain is a PID if its set of primes is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization.over_prime.mem_normalized_factors_of_is_prime [decidable_eq (ideal Sₚ)]
{P : ideal Sₚ} (hP : is_prime P) (hP0 : P ≠ ⊥) :
P ∈ normalized_factors (ideal.map (algebra_map R Sₚ) p) | begin
have non_zero_div : algebra.algebra_map_submonoid S p.prime_compl ≤ S⁰ :=
map_le_non_zero_divisors_of_injective _ (no_zero_smul_divisors.algebra_map_injective _ _)
p.prime_compl_le_non_zero_divisors,
letI : algebra (localization.at_prime p) Sₚ := localization_algebra p.prime_compl S,
haveI : is_sc... | lemma | is_localization.over_prime.mem_normalized_factors_of_is_prime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/pid.lean | [
"ring_theory.dedekind_domain.dvr",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra",
"algebra.algebra_map_submonoid",
"algebra.is_integral",
"algebra_map",
"discrete_valuation_ring.iff_pid_with_one_nonzero_prime",
"ideal",
"ideal.comap_is_prime",
"ideal.eq_bot_of_comap_eq_bot",
"ideal.map",
"ideal.map_eq_bot_iff_of_injective",
"ideal.map_le_iff_le_comap",
"ideal.map... | If `p` is a prime in the Dedekind domain `R`, `S` an extension of `R` and `Sₚ` the localization
of `S` at `p`, then all primes in `Sₚ` are factors of the image of `p` in `Sₚ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain.is_principal_ideal_ring_localization_over_prime :
is_principal_ideal_ring Sₚ | begin
letI := classical.dec_eq (ideal Sₚ),
letI := classical.dec_pred (λ (P : ideal Sₚ), P.is_prime),
refine is_principal_ideal_ring.of_finite_primes
(set.finite.of_finset (finset.filter (λ P, P.is_prime)
({⊥} ∪ (normalized_factors (ideal.map (algebra_map R Sₚ) p)).to_finset))
(λ P, _)),
rw [fin... | theorem | is_dedekind_domain.is_principal_ideal_ring_localization_over_prime | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/pid.lean | [
"ring_theory.dedekind_domain.dvr",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra_map",
"and_iff_right_of_imp",
"classical.dec_eq",
"classical.dec_pred",
"finset.filter",
"finset.mem_filter",
"finset.mem_singleton",
"finset.mem_union",
"ideal",
"ideal.map",
"is_localization.over_prime.mem_normalized_factors_of_is_prime",
"is_principal_ideal_ring",
"is_principal_i... | Let `p` be a prime in the Dedekind domain `R` and `S` be an integral extension of `R`,
then the localization `Sₚ` of `S` at `p` is a PID. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_of_ne_zero_to_fun (x : Kˣ) : multiplicative ℤ | let hx := is_localization.sec R⁰ (x : K) in multiplicative.of_add $
(-(associates.mk v.as_ideal).count (associates.mk $ ideal.span {hx.fst}).factors : ℤ)
- (-(associates.mk v.as_ideal).count (associates.mk $ ideal.span {(hx.snd : R)}).factors : ℤ) | def | is_dedekind_domain.height_one_spectrum.valuation_of_ne_zero_to_fun | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"associates.mk",
"ideal.span",
"is_localization.sec",
"multiplicative",
"multiplicative.of_add"
] | The multiplicative `v`-adic valuation on `Kˣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_of_ne_zero_to_fun_eq (x : Kˣ) :
(v.valuation_of_ne_zero_to_fun x : ℤₘ₀) = v.valuation (x : K) | begin
change _ = _ * _,
rw [units.coe_inv],
change _ = ite _ _ _ * (ite (coe _ = _) _ _)⁻¹,
rw [is_localization.to_localization_map_sec,
if_neg $ is_localization.sec_fst_ne_zero le_rfl x.ne_zero,
if_neg $ non_zero_divisors.coe_ne_zero _],
any_goals { exact is_domain.to_nontrivial R },
refl
end | lemma | is_dedekind_domain.height_one_spectrum.valuation_of_ne_zero_to_fun_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"is_localization.sec_fst_ne_zero",
"is_localization.to_localization_map_sec",
"le_rfl",
"non_zero_divisors.coe_ne_zero",
"units.coe_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_of_ne_zero : Kˣ →* multiplicative ℤ | { to_fun := v.valuation_of_ne_zero_to_fun,
map_one' := by { rw [← with_zero.coe_inj, valuation_of_ne_zero_to_fun_eq], exact map_one _ },
map_mul' := λ _ _, by { rw [← with_zero.coe_inj, with_zero.coe_mul],
simp only [valuation_of_ne_zero_to_fun_eq], exact map_mul _ _ _ } } | def | is_dedekind_domain.height_one_spectrum.valuation_of_ne_zero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"map_mul",
"map_one",
"multiplicative",
"with_zero.coe_mul"
] | The multiplicative `v`-adic valuation on `Kˣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_of_ne_zero_eq (x : Kˣ) :
(v.valuation_of_ne_zero x : ℤₘ₀) = v.valuation (x : K) | valuation_of_ne_zero_to_fun_eq v x | lemma | is_dedekind_domain.height_one_spectrum.valuation_of_ne_zero_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_of_unit_eq (x : Rˣ) :
v.valuation_of_ne_zero (units.map (algebra_map R K : R →* K) x) = 1 | begin
rw [← with_zero.coe_inj, valuation_of_ne_zero_eq, units.coe_map, eq_iff_le_not_lt],
split,
{ exact v.valuation_le_one x },
{ cases x with x _ hx _,
change ¬v.valuation (algebra_map R K x) < 1,
apply_fun v.int_valuation at hx,
rw [map_one, map_mul] at hx,
rw [not_lt, ← hx, ← mul_one $ v.val... | lemma | is_dedekind_domain.height_one_spectrum.valuation_of_unit_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"algebra_map",
"eq_iff_le_not_lt",
"left_ne_zero_of_mul_eq_one",
"map_mul",
"map_one",
"mul_le_mul_left₀",
"mul_one",
"units.coe_map",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_of_ne_zero_mod (n : ℕ) : K/n →* multiplicative (zmod n) | (int.quotient_zmultiples_nat_equiv_zmod n).to_multiplicative.to_monoid_hom.comp $
quotient_group.map (pow_monoid_hom n : Kˣ →* Kˣ).range
(add_subgroup.zmultiples (n : ℤ)).to_subgroup v.valuation_of_ne_zero
begin
rintro _ ⟨x, rfl⟩,
exact ⟨v.valuation_of_ne_zero x, by simpa only [pow_monoid_hom_apply, map_pow, in... | def | is_dedekind_domain.height_one_spectrum.valuation_of_ne_zero_mod | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"add_subgroup.zmultiples",
"int.quotient_zmultiples_nat_equiv_zmod",
"int.to_add_pow",
"map_pow",
"multiplicative",
"pow_monoid_hom",
"quotient_group.map",
"zmod"
] | The multiplicative `v`-adic valuation on `Kˣ` modulo `n`-th powers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_of_unit_mod_eq (n : ℕ) (x : Rˣ) :
v.valuation_of_ne_zero_mod n (units.map (algebra_map R K : R →* K) x : K/n) = 1 | by rw [valuation_of_ne_zero_mod, monoid_hom.comp_apply, ← quotient_group.coe_mk',
quotient_group.map_mk', valuation_of_unit_eq, quotient_group.coe_one, map_one] | lemma | is_dedekind_domain.height_one_spectrum.valuation_of_unit_mod_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"algebra_map",
"map_one",
"monoid_hom.comp_apply",
"quotient_group.coe_mk'",
"quotient_group.coe_one",
"quotient_group.map_mk'",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
selmer_group : subgroup $ K/n | { carrier := {x : K/n | ∀ v ∉ S, (v : height_one_spectrum R).valuation_of_ne_zero_mod n x = 1},
one_mem' := λ _ _, by rw [map_one],
mul_mem' := λ _ _ hx hy v hv, by rw [map_mul, hx v hv, hy v hv, one_mul],
inv_mem' := λ _ hx v hv, by rw [map_inv, hx v hv, inv_one] } | def | is_dedekind_domain.selmer_group | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"inv_one",
"map_inv",
"map_mul",
"map_one",
"one_mul",
"subgroup"
] | The Selmer group `K⟮S, n⟯`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone (hS : S ≤ S') : K⟮S, n⟯ ≤ (K⟮S', n⟯) | λ _ hx v, hx v ∘ mt (@hS v) | lemma | is_dedekind_domain.selmer_group.monotone | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation : K⟮S, n⟯ →* S → multiplicative (zmod n) | { to_fun := λ x v, (v : height_one_spectrum R).valuation_of_ne_zero_mod n (x : K/n),
map_one' := funext $ λ v, map_one _,
map_mul' := λ x y, funext $ λ v, map_mul _ x y } | def | is_dedekind_domain.selmer_group.valuation | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"map_mul",
"map_one",
"multiplicative",
"valuation",
"zmod"
] | The multiplicative `v`-adic valuations on `K⟮S, n⟯` for all `v ∈ S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_ker_eq :
valuation.ker = (K⟮(∅ : set $ height_one_spectrum R), n⟯).subgroup_of (K⟮S, n⟯) | begin
ext ⟨_, hx⟩,
split,
{ intros hx' v _,
by_cases hv : v ∈ S,
{ exact congr_fun hx' ⟨v, hv⟩ },
{ exact hx v hv } },
{ exact λ hx', funext $ λ v, hx' v $ set.not_mem_empty v }
end | lemma | is_dedekind_domain.selmer_group.valuation_ker_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"set.not_mem_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_unit {n : ℕ} : Rˣ →* K⟮(∅ : set $ height_one_spectrum R), n⟯ | { to_fun := λ x, ⟨quotient_group.mk $ units.map (algebra_map R K).to_monoid_hom x,
λ v _, v.valuation_of_unit_mod_eq n x⟩,
map_one' := by simpa only [map_one],
map_mul' := λ _ _, by simpa only [map_mul] } | def | is_dedekind_domain.selmer_group.from_unit | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"algebra_map",
"map_mul",
"map_one",
"units.map"
] | The natural homomorphism from `Rˣ` to `K⟮∅, n⟯`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_unit_ker [hn : fact $ 0 < n] :
(@from_unit R _ _ _ K _ _ _ n).ker = (pow_monoid_hom n : Rˣ →* Rˣ).range | begin
ext ⟨_, _, _, _⟩,
split,
{ intro hx,
rcases (quotient_group.eq_one_iff _).mp (subtype.mk.inj hx) with ⟨⟨v, i, vi, iv⟩, hx⟩,
have hv : ↑(_ ^ n : Kˣ) = algebra_map R K _ := congr_arg units.val hx,
have hi : ↑(_ ^ n : Kˣ)⁻¹ = algebra_map R K _ := congr_arg units.inv hx,
rw [units.coe_pow] at hv... | lemma | is_dedekind_domain.selmer_group.from_unit_ker | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"algebra_map",
"fact",
"inv_pow",
"is_integral_algebra_map",
"is_integrally_closed.exists_algebra_map_eq_of_is_integral_pow",
"map_eq_one_iff",
"map_mul",
"map_pow",
"no_zero_smul_divisors.algebra_map_injective",
"pow_monoid_hom",
"quotient_group.eq_one_iff",
"units.coe_mk",
"units.coe_pow",... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
from_unit_lift [fact $ 0 < n] : R/n →* K⟮(∅ : set $ height_one_spectrum R), n⟯ | (quotient_group.ker_lift _).comp
(quotient_group.quotient_mul_equiv_of_eq from_unit_ker).symm.to_monoid_hom | def | is_dedekind_domain.selmer_group.from_unit_lift | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"fact",
"quotient_group.ker_lift",
"quotient_group.quotient_mul_equiv_of_eq"
] | The injection induced by the natural homomorphism from `Rˣ` to `K⟮∅, n⟯`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
from_unit_lift_injective [fact $ 0 < n] :
function.injective $ @from_unit_lift R _ _ _ K _ _ _ n _ | function.injective.comp (quotient_group.ker_lift_injective _) (mul_equiv.injective _) | lemma | is_dedekind_domain.selmer_group.from_unit_lift_injective | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/selmer_group.lean | [
"algebra.hom.equiv.type_tags",
"data.zmod.quotient",
"ring_theory.dedekind_domain.adic_valuation",
"ring_theory.norm"
] | [
"fact",
"mul_equiv.injective",
"quotient_group.ker_lift_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integer : subalgebra R K | { algebra_map_mem' := λ x v _, v.valuation_le_one x,
.. (⨅ v ∉ S, (v : height_one_spectrum R).valuation.valuation_subring.to_subring).copy
{x : K | ∀ v ∉ S, (v : height_one_spectrum R).valuation x ≤ 1} $ set.ext $ λ _,
by simpa only [set_like.mem_coe, subring.mem_infi] } | def | set.integer | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/S_integer.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"set.ext",
"set_like.mem_coe",
"subalgebra",
"subring.mem_infi",
"valuation"
] | The `R`-subalgebra of `S`-integers of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integer_eq :
(S.integer K).to_subring
= ⨅ v ∉ S, (v : height_one_spectrum R).valuation.valuation_subring.to_subring | set_like.ext' $ by simpa only [integer, subring.copy_eq] | lemma | set.integer_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/S_integer.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"set_like.ext'",
"subring.copy_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integer_valuation_le_one (x : S.integer K) {v : height_one_spectrum R} (hv : v ∉ S) :
v.valuation (x : K) ≤ 1 | x.property v hv | lemma | set.integer_valuation_le_one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/S_integer.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit : subgroup Kˣ | (⨅ v ∉ S, (v : height_one_spectrum R).valuation.valuation_subring.unit_group).copy
{x : Kˣ | ∀ v ∉ S, (v : height_one_spectrum R).valuation (x : K) = 1} $ set.ext $ λ _,
by simpa only [set_like.mem_coe, subgroup.mem_infi, valuation.mem_unit_group_iff] | def | set.unit | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/S_integer.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"set.ext",
"set_like.mem_coe",
"subgroup",
"subgroup.mem_infi",
"valuation",
"valuation.mem_unit_group_iff"
] | The subgroup of `S`-units of `Kˣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_eq :
S.unit K = ⨅ v ∉ S, (v : height_one_spectrum R).valuation.valuation_subring.unit_group | subgroup.copy_eq _ _ _ | lemma | set.unit_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/S_integer.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"subgroup.copy_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_valuation_eq_one (x : S.unit K) {v : height_one_spectrum R} (hv : v ∉ S) :
v.valuation (x : K) = 1 | x.property v hv | lemma | set.unit_valuation_eq_one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/S_integer.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_equiv_units_integer : S.unit K ≃* (S.integer K)ˣ | { to_fun := λ x, ⟨⟨x, λ v hv, (x.property v hv).le⟩, ⟨↑x⁻¹, λ v hv, ((x⁻¹).property v hv).le⟩,
subtype.ext x.val.val_inv, subtype.ext x.val.inv_val⟩,
inv_fun := λ x, ⟨units.mk0 x $ λ hx, x.ne_zero ((subring.coe_eq_zero_iff _).mp hx),
λ v hv, eq_one_of_one_le_mul_left (x.val.property v hv) (x.inv.property... | def | set.unit_equiv_units_integer | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/S_integer.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"eq.ge",
"eq_one_of_one_le_mul_left",
"inv_fun",
"map_mul",
"subring.coe_eq_zero_iff",
"subtype.ext"
] | The group of `S`-units is the group of units of the ring of `S`-integers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
derivation (R : Type*) (A : Type*) [comm_semiring R] [comm_semiring A]
[algebra R A] (M : Type*) [add_comm_monoid M] [module A M] [module R M]
extends A →ₗ[R] M | (map_one_eq_zero' : to_linear_map 1 = 0)
(leibniz' (a b : A) : to_linear_map (a * b) = a • to_linear_map b + b • to_linear_map a) | structure | derivation | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"add_comm_monoid",
"algebra",
"comm_semiring",
"module"
] | `D : derivation R A M` is an `R`-linear map from `A` to `M` that satisfies the `leibniz`
equality. We also require that `D 1 = 0`. See `derivation.mk'` for a constructor that deduces this
assumption from the Leibniz rule when `M` is cancellative.
TODO: update this when bimodules are defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe : D.to_fun = ⇑D | rfl | lemma | derivation.to_fun_eq_coe | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_to_linear_map : has_coe (derivation R A M) (A →ₗ[R] M) | ⟨λ D, D.to_linear_map⟩ | instance | derivation.has_coe_to_linear_map | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_eq_coe : D.to_linear_map = D | rfl | lemma | derivation.to_linear_map_eq_coe | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : A →ₗ[R] M) (h₁ h₂) :
((⟨f, h₁, h₂⟩ : derivation R A M) : A → M) = f | rfl | lemma | derivation.mk_coe | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_coe (f : derivation R A M) : ⇑(f : A →ₗ[R] M) = f | rfl | lemma | derivation.coe_fn_coe | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @function.injective (derivation R A M) (A → M) coe_fn | fun_like.coe_injective | lemma | derivation.coe_injective | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation",
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (H : ∀ a, D1 a = D2 a) : D1 = D2 | fun_like.ext _ _ H | theorem | derivation.ext | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun (h : D1 = D2) (a : A) : D1 a = D2 a | fun_like.congr_fun h a | lemma | derivation.congr_fun | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add : D (a + b) = D a + D b | map_add D a b | lemma | derivation.map_add | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : D 0 = 0 | map_zero D | lemma | derivation.map_zero | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul : D (r • a) = r • D a | D.to_linear_map.map_smul r a | lemma | derivation.map_smul | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leibniz : D (a * b) = a • D b + b • D a | D.leibniz' _ _ | lemma | derivation.leibniz | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sum {ι : Type*} (s : finset ι) (f : ι → A) : D (∑ i in s, f i) = ∑ i in s, D (f i) | D.to_linear_map.map_sum | lemma | derivation.map_sum | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul_of_tower {S : Type*} [has_smul S A] [has_smul S M]
[linear_map.compatible_smul A M S R] (D : derivation R A M) (r : S) (a : A) :
D (r • a) = r • D a | D.to_linear_map.map_smul_of_tower r a | lemma | derivation.map_smul_of_tower | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation",
"has_smul",
"linear_map.compatible_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one_eq_zero : D 1 = 0 | D.map_one_eq_zero' | lemma | derivation.map_one_eq_zero | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_algebra_map : D (algebra_map R A r) = 0 | by rw [←mul_one r, ring_hom.map_mul, ring_hom.map_one, ←smul_def, map_smul, map_one_eq_zero,
smul_zero] | lemma | derivation.map_algebra_map | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"algebra_map",
"ring_hom.map_mul",
"ring_hom.map_one",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_nat (n : ℕ) : D (n : A) = 0 | by rw [← nsmul_one, D.map_smul_of_tower n, map_one_eq_zero, smul_zero] | lemma | derivation.map_coe_nat | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"nsmul_one",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leibniz_pow (n : ℕ) : D (a ^ n) = n • a ^ (n - 1) • D a | begin
induction n with n ihn,
{ rw [pow_zero, map_one_eq_zero, zero_smul] },
{ rcases (zero_le n).eq_or_lt with (rfl|hpos),
{ rw [pow_one, one_smul, pow_zero, one_smul] },
{ have : a * a ^ (n - 1) = a ^ n, by rw [← pow_succ, nat.sub_add_cancel hpos],
simp only [pow_succ, leibniz, ihn, smul_comm a n,... | lemma | derivation.leibniz_pow | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"add_smul",
"nat.add_succ_sub_one",
"one_smul",
"pow_one",
"pow_succ",
"pow_zero",
"smul_smul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_adjoin {s : set A} (h : set.eq_on D1 D2 s) : set.eq_on D1 D2 (adjoin R s) | λ x hx, algebra.adjoin_induction hx h
(λ r, (D1.map_algebra_map r).trans (D2.map_algebra_map r).symm)
(λ x y hx hy, by simp only [map_add, *])
(λ x y hx hy, by simp only [leibniz, *]) | lemma | derivation.eq_on_adjoin | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"algebra.adjoin_induction",
"set.eq_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_of_adjoin_eq_top (s : set A) (hs : adjoin R s = ⊤) (h : set.eq_on D1 D2 s) : D1 = D2 | ext $ λ a, eq_on_adjoin h $ hs.symm ▸ trivial | lemma | derivation.ext_of_adjoin_eq_top | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"set.eq_on"
] | If adjoin of a set is the whole algebra, then any two derivations equal on this set are equal
on the whole algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_zero : ⇑(0 : derivation R A M) = 0 | rfl | lemma | derivation.coe_zero | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero_linear_map : ↑(0 : derivation R A M) = (0 : A →ₗ[R] M) | rfl | lemma | derivation.coe_zero_linear_map | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (a : A) : (0 : derivation R A M) a = 0 | rfl | lemma | derivation.zero_apply | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (D1 D2 : derivation R A M) : ⇑(D1 + D2) = D1 + D2 | rfl | lemma | derivation.coe_add | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_linear_map (D1 D2 : derivation R A M) : ↑(D1 + D2) = (D1 + D2 : A →ₗ[R] M) | rfl | lemma | derivation.coe_add_linear_map | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply : (D1 + D2) a = D1 a + D2 a | rfl | lemma | derivation.add_apply | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (r : S) (D : derivation R A M) : ⇑(r • D) = r • D | rfl | lemma | derivation.coe_smul | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul_linear_map (r : S) (D : derivation R A M) :
↑(r • D) = (r • D : A →ₗ[R] M) | rfl | lemma | derivation.coe_smul_linear_map | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (r : S) (D : derivation R A M) : (r • D) a = r • D a | rfl | lemma | derivation.smul_apply | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_add_monoid_hom : derivation R A M →+ (A → M) | { to_fun := coe_fn, map_zero' := coe_zero, map_add' := coe_add } | def | derivation.coe_fn_add_monoid_hom | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | `coe_fn` as an `add_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.linear_map.comp_der : derivation R A M →ₗ[R] derivation R A N | { to_fun := λ D,
{ to_linear_map := (f : M →ₗ[R] N).comp (D : A →ₗ[R] M),
map_one_eq_zero' := by simp only [linear_map.comp_apply, coe_fn_coe, map_one_eq_zero, map_zero],
leibniz' := λ a b, by simp only [coe_fn_coe, linear_map.comp_apply, linear_map.map_add,
leibniz, linear_map.coe_coe_is_scalar_tow... | def | linear_map.comp_der | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation",
"linear_map.coe_coe_is_scalar_tower",
"linear_map.comp_apply",
"linear_map.map_add",
"linear_map.map_smul"
] | We can push forward derivations using linear maps, i.e., the composition of a derivation with a
linear map is a derivation. Furthermore, this operation is linear on the spaces of derivations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_linear_map_comp :
(f.comp_der D : A →ₗ[R] N) = (f : M →ₗ[R] N).comp (D : A →ₗ[R] M) | rfl | lemma | derivation.coe_to_linear_map_comp | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp :
(f.comp_der D : A → N) = (f : M →ₗ[R] N).comp (D : A →ₗ[R] M) | rfl | lemma | derivation.coe_comp | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
llcomp : (M →ₗ[A] N) →ₗ[A] derivation R A M →ₗ[R] derivation R A N | { to_fun := λ f, f.comp_der,
map_add' := λ f₁ f₂, by { ext, refl },
map_smul' := λ r D, by { ext, refl } } | def | derivation.llcomp | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | The composition of a derivation with a linear map as a bilinear map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.linear_equiv.comp_der : derivation R A M ≃ₗ[R] derivation R A N | { inv_fun := e.symm.to_linear_map.comp_der,
left_inv := λ D, by { ext a, exact e.symm_apply_apply (D a) },
right_inv := λ D, by { ext a, exact e.apply_symm_apply (D a) },
..e.to_linear_map.comp_der } | def | linear_equiv.comp_der | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation",
"inv_fun"
] | Pushing a derivation foward through a linear equivalence is an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars (d : derivation S A M) : derivation R A M | { map_one_eq_zero' := d.map_one_eq_zero,
leibniz' := d.leibniz,
to_linear_map := d.to_linear_map.restrict_scalars R } | def | derivation.restrict_scalars | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation",
"restrict_scalars"
] | If `A` is both an `R`-algebra and an `S`-algebra; `M` is both an `R`-module and an `S`-module,
then an `S`-derivation `A → M` is also an `R`-derivation if it is also `R`-linear. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk' (D : A →ₗ[R] M) (h : ∀ a b, D (a * b) = a • D b + b • D a) : derivation R A M | { to_linear_map := D,
map_one_eq_zero' := add_right_eq_self.1 $ by simpa only [one_smul, one_mul] using (h 1 1).symm,
leibniz' := h } | def | derivation.mk' | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation",
"mk'",
"one_mul",
"one_smul"
] | Define `derivation R A M` from a linear map when `M` is cancellative by verifying the Leibniz
rule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk' (D : A →ₗ[R] M) (h) : ⇑(mk' D h) = D | rfl | lemma | derivation.coe_mk' | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk'_linear_map (D : A →ₗ[R] M) (h) : (mk' D h : A →ₗ[R] M) = D | rfl | lemma | derivation.coe_mk'_linear_map | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg : D (-a) = -D a | map_neg D a | lemma | derivation.map_neg | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub : D (a - b) = D a - D b | map_sub D a b | lemma | derivation.map_sub | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_int (n : ℤ) : D (n : A) = 0 | by rw [← zsmul_one, D.map_smul_of_tower n, map_one_eq_zero, smul_zero] | lemma | derivation.map_coe_int | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"smul_zero",
"zsmul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leibniz_of_mul_eq_one {a b : A} (h : a * b = 1) : D a = -a^2 • D b | begin
rw neg_smul,
refine eq_neg_of_add_eq_zero_left _,
calc D a + a ^ 2 • D b = a • b • D a + a • a • D b : by simp only [smul_smul, h, one_smul, sq]
... = a • D (a * b) : by rw [leibniz, smul_add, add_comm]
... = 0 : by rw [h, map_one... | lemma | derivation.leibniz_of_mul_eq_one | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"neg_smul",
"one_smul",
"smul_add",
"smul_smul",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leibniz_inv_of [invertible a] : D (⅟a) = -⅟a^2 • D a | D.leibniz_of_mul_eq_one $ inv_of_mul_self a | lemma | derivation.leibniz_inv_of | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"inv_of_mul_self",
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leibniz_inv {K : Type*} [field K] [module K M] [algebra R K] (D : derivation R K M) (a : K) :
D (a⁻¹) = -a⁻¹ ^ 2 • D a | begin
rcases eq_or_ne a 0 with (rfl|ha),
{ simp },
{ exact D.leibniz_of_mul_eq_one (inv_mul_cancel ha) }
end | lemma | derivation.leibniz_inv | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"algebra",
"derivation",
"eq_or_ne",
"field",
"inv_mul_cancel",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (D : derivation R A M) : ⇑(-D) = -D | rfl | lemma | derivation.coe_neg | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg_linear_map (D : derivation R A M) : ↑(-D) = (-D : A →ₗ[R] M) | rfl | lemma | derivation.coe_neg_linear_map | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_apply : (-D) a = -D a | rfl | lemma | derivation.neg_apply | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (D1 D2 : derivation R A M) : ⇑(D1 - D2) = D1 - D2 | rfl | lemma | derivation.coe_sub | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub_linear_map (D1 D2 : derivation R A M) : ↑(D1 - D2) = (D1 - D2 : A →ₗ[R] M) | rfl | lemma | derivation.coe_sub_linear_map | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [
"derivation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply : (D1 - D2) a = D1 a - D2 a | rfl | lemma | derivation.sub_apply | ring_theory.derivation | src/ring_theory/derivation/basic.lean | [
"ring_theory.adjoin.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutator_coe_linear_map :
↑⁅D1, D2⁆ = ⁅(D1 : module.End R A), (D2 : module.End R A)⁆ | rfl | lemma | derivation.commutator_coe_linear_map | ring_theory.derivation | src/ring_theory/derivation/lie.lean | [
"algebra.lie.of_associative",
"ring_theory.derivation.basic"
] | [
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutator_apply : ⁅D1, D2⁆ a = D1 (D2 a) - D2 (D1 a) | rfl | lemma | derivation.commutator_apply | ring_theory.derivation | src/ring_theory/derivation/lie.lean | [
"algebra.lie.of_associative",
"ring_theory.derivation.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diff_to_ideal_of_quotient_comp_eq (f₁ f₂ : A →ₐ[R] B)
(e : (ideal.quotient.mkₐ R I).comp f₁ = (ideal.quotient.mkₐ R I).comp f₂) :
A →ₗ[R] I | linear_map.cod_restrict (I.restrict_scalars _) (f₁.to_linear_map - f₂.to_linear_map)
begin
intro x,
change f₁ x - f₂ x ∈ I,
rw [← ideal.quotient.eq, ← ideal.quotient.mkₐ_eq_mk R, ← alg_hom.comp_apply, e],
refl,
end | def | diff_to_ideal_of_quotient_comp_eq | ring_theory.derivation | src/ring_theory/derivation/to_square_zero.lean | [
"ring_theory.derivation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"alg_hom.comp_apply",
"ideal.quotient.eq",
"ideal.quotient.mkₐ",
"ideal.quotient.mkₐ_eq_mk",
"linear_map.cod_restrict"
] | If `f₁ f₂ : A →ₐ[R] B` are two lifts of the same `A →ₐ[R] B ⧸ I`,
we may define a map `f₁ - f₂ : A →ₗ[R] I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diff_to_ideal_of_quotient_comp_eq_apply (f₁ f₂ : A →ₐ[R] B)
(e : (ideal.quotient.mkₐ R I).comp f₁ = (ideal.quotient.mkₐ R I).comp f₂) (x : A) :
((diff_to_ideal_of_quotient_comp_eq I f₁ f₂ e) x : B) = f₁ x - f₂ x | rfl | lemma | diff_to_ideal_of_quotient_comp_eq_apply | ring_theory.derivation | src/ring_theory/derivation/to_square_zero.lean | [
"ring_theory.derivation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"diff_to_ideal_of_quotient_comp_eq",
"ideal.quotient.mkₐ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivation_to_square_zero_of_lift
(f : A →ₐ[R] B) (e : (ideal.quotient.mkₐ R I).comp f = is_scalar_tower.to_alg_hom R A (B ⧸ I)) :
derivation R A I | begin
refine
{ map_one_eq_zero' := _,
leibniz' := _,
..(diff_to_ideal_of_quotient_comp_eq I f (is_scalar_tower.to_alg_hom R A B) _) },
{ rw e, ext, refl },
{ ext, change f 1 - algebra_map A B 1 = 0, rw [map_one, map_one, sub_self] },
{ intros x y,
let F := diff_to_ideal_of_quotient_comp_eq I f (is... | def | derivation_to_square_zero_of_lift | ring_theory.derivation | src/ring_theory/derivation/to_square_zero.lean | [
"ring_theory.derivation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"algebra.smul_def",
"algebra_map",
"derivation",
"diff_to_ideal_of_quotient_comp_eq",
"diff_to_ideal_of_quotient_comp_eq_apply",
"ideal.mem_bot",
"ideal.mul_mem_mul",
"ideal.quotient.mkₐ",
"is_scalar_tower.coe_to_alg_hom'",
"is_scalar_tower.to_alg_hom",
"linear_map.coe_mk",
"linear_map.to_fun_... | Given a tower of algebras `R → A → B`, and a square-zero `I : ideal B`, each lift `A →ₐ[R] B`
of the canonical map `A →ₐ[R] B ⧸ I` corresponds to a `R`-derivation from `A` to `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
derivation_to_square_zero_of_lift_apply (f : A →ₐ[R] B)
(e : (ideal.quotient.mkₐ R I).comp f = is_scalar_tower.to_alg_hom R A (B ⧸ I))
(x : A) : (derivation_to_square_zero_of_lift I hI f e x : B) = f x - algebra_map A B x | rfl | lemma | derivation_to_square_zero_of_lift_apply | ring_theory.derivation | src/ring_theory/derivation/to_square_zero.lean | [
"ring_theory.derivation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"algebra_map",
"derivation_to_square_zero_of_lift",
"ideal.quotient.mkₐ",
"is_scalar_tower.to_alg_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of_derivation_to_square_zero (f : derivation R A I) :
A →ₐ[R] B | { to_fun := λ x, f x + algebra_map A B x,
map_one' := by rw [map_one, f.map_one_eq_zero, submodule.coe_zero, zero_add],
map_mul' := λ x y, begin
have : (f x : B) * (f y) = 0,
{ rw [← ideal.mem_bot, ← hI, pow_two], convert (ideal.mul_mem_mul (f x).2 (f y).2) using 1 },
simp only [map_mul, f.leibniz, add_... | def | lift_of_derivation_to_square_zero | ring_theory.derivation | src/ring_theory/derivation/to_square_zero.lean | [
"ring_theory.derivation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"algebra.smul_def",
"algebra_map",
"derivation",
"derivation.map_algebra_map",
"ideal.mem_bot",
"ideal.mul_mem_mul",
"is_scalar_tower.to_alg_hom",
"map_mul",
"map_one",
"pow_two",
"ring",
"submodule.coe_add",
"submodule.coe_smul_of_tower",
"submodule.coe_zero"
] | Given a tower of algebras `R → A → B`, and a square-zero `I : ideal B`, each `R`-derivation
from `A` to `I` corresponds to a lift `A →ₐ[R] B` of the canonical map `A →ₐ[R] B ⧸ I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_of_derivation_to_square_zero_mk_apply (d : derivation R A I) (x : A) :
ideal.quotient.mk I (lift_of_derivation_to_square_zero I hI d x) = algebra_map A (B ⧸ I) x | by { rw [lift_of_derivation_to_square_zero_apply, map_add,
ideal.quotient.eq_zero_iff_mem.mpr (d x).prop, zero_add], refl } | lemma | lift_of_derivation_to_square_zero_mk_apply | ring_theory.derivation | src/ring_theory/derivation/to_square_zero.lean | [
"ring_theory.derivation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"algebra_map",
"derivation",
"ideal.quotient.mk",
"lift_of_derivation_to_square_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.