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 |
|---|---|---|---|---|---|---|---|---|---|---|
int_valuation_def_if_neg {r : R} (hr : r ≠ 0) : v.int_valuation_def r = (multiplicative.of_add
(-(associates.mk v.as_ideal).count (associates.mk (ideal.span {r} : ideal R)).factors : ℤ)) | if_neg hr | lemma | is_dedekind_domain.height_one_spectrum.int_valuation_def_if_neg | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"associates.mk",
"ideal",
"ideal.span",
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_valuation_ne_zero (x : R) (hx : x ≠ 0) : v.int_valuation_def x ≠ 0 | begin
rw [int_valuation_def, if_neg hx],
exact with_zero.coe_ne_zero,
end | lemma | is_dedekind_domain.height_one_spectrum.int_valuation_ne_zero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [] | Nonzero elements have nonzero adic valuation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation_ne_zero' (x : non_zero_divisors R) : v.int_valuation_def x ≠ 0 | v.int_valuation_ne_zero x (non_zero_divisors.coe_ne_zero x) | lemma | is_dedekind_domain.height_one_spectrum.int_valuation_ne_zero' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"non_zero_divisors",
"non_zero_divisors.coe_ne_zero"
] | Nonzero divisors have nonzero valuation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation_zero_le (x : non_zero_divisors R) : 0 < v.int_valuation_def x | begin
rw [v.int_valuation_def_if_neg (non_zero_divisors.coe_ne_zero x)],
exact with_zero.zero_lt_coe _,
end | lemma | is_dedekind_domain.height_one_spectrum.int_valuation_zero_le | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"non_zero_divisors",
"non_zero_divisors.coe_ne_zero",
"with_zero.zero_lt_coe"
] | Nonzero divisors have valuation greater than zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation_le_one (x : R) : v.int_valuation_def x ≤ 1 | begin
rw int_valuation_def,
by_cases hx : x = 0,
{ rw if_pos hx, exact with_zero.zero_le 1 },
{ rw [if_neg hx, ← with_zero.coe_one, ← of_add_zero, with_zero.coe_le_coe, of_add_le,
right.neg_nonpos_iff],
exact int.coe_nat_nonneg _ }
end | lemma | is_dedekind_domain.height_one_spectrum.int_valuation_le_one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"int.coe_nat_nonneg",
"of_add_zero",
"with_zero.coe_le_coe",
"with_zero.coe_one",
"with_zero.zero_le"
] | The `v`-adic valuation on `R` is bounded above by 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation_lt_one_iff_dvd (r : R) :
v.int_valuation_def r < 1 ↔ v.as_ideal ∣ ideal.span {r} | begin
rw int_valuation_def,
split_ifs with hr,
{ simpa [hr] using (with_zero.zero_lt_coe _) },
{ rw [← with_zero.coe_one, ← of_add_zero, with_zero.coe_lt_coe, of_add_lt, neg_lt_zero,
← int.coe_nat_zero, int.coe_nat_lt, zero_lt_iff],
have h : (ideal.span {r} : ideal R) ≠ 0,
{ rw [ne.def, ideal.zero... | lemma | is_dedekind_domain.height_one_spectrum.int_valuation_lt_one_iff_dvd | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"associates.count_ne_zero_iff_dvd",
"ideal",
"ideal.span",
"ideal.span_singleton_eq_bot",
"ideal.zero_eq_bot",
"int.coe_nat_lt",
"of_add_zero",
"with_zero.coe_lt_coe",
"with_zero.coe_one",
"with_zero.zero_lt_coe",
"zero_lt_iff"
] | The `v`-adic valuation of `r ∈ R` is less than 1 if and only if `v` divides the ideal `(r)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation_le_pow_iff_dvd (r : R) (n : ℕ) :
v.int_valuation_def r ≤ multiplicative.of_add (-(n : ℤ)) ↔ v.as_ideal^n ∣ ideal.span {r} | begin
rw int_valuation_def,
split_ifs with hr,
{ simp_rw [hr, ideal.dvd_span_singleton, zero_le', submodule.zero_mem], },
{ rw [with_zero.coe_le_coe, of_add_le, neg_le_neg_iff, int.coe_nat_le, ideal.dvd_span_singleton,
← associates.le_singleton_iff, associates.prime_pow_dvd_iff_le (associates.mk_ne_zero'.... | lemma | is_dedekind_domain.height_one_spectrum.int_valuation_le_pow_iff_dvd | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"associates.le_singleton_iff",
"associates.prime_pow_dvd_iff_le",
"ideal.dvd_span_singleton",
"ideal.span",
"int.coe_nat_le",
"multiplicative.of_add",
"submodule.zero_mem",
"with_zero.coe_le_coe",
"zero_le'"
] | The `v`-adic valuation of `r ∈ R` is less than `multiplicative.of_add (-n)` if and only if
`vⁿ` divides the ideal `(r)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation.map_zero' : v.int_valuation_def 0 = 0 | v.int_valuation_def_if_pos (eq.refl 0) | lemma | is_dedekind_domain.height_one_spectrum.int_valuation.map_zero' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [] | The `v`-adic valuation of `0 : R` equals 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation.map_one' : v.int_valuation_def 1 = 1 | by rw [v.int_valuation_def_if_neg (zero_ne_one.symm : (1 : R) ≠ 0), ideal.span_singleton_one,
← ideal.one_eq_top, associates.mk_one, associates.factors_one, associates.count_zero
(by apply v.associates_irreducible), int.coe_nat_zero, neg_zero, of_add_zero, with_zero.coe_one] | lemma | is_dedekind_domain.height_one_spectrum.int_valuation.map_one' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"associates.count_zero",
"associates.factors_one",
"associates.mk_one",
"ideal.one_eq_top",
"ideal.span_singleton_one",
"of_add_zero",
"with_zero.coe_one"
] | The `v`-adic valuation of `1 : R` equals 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation.map_mul' (x y : R) :
v.int_valuation_def (x * y) = v.int_valuation_def x * v.int_valuation_def y | begin
simp only [int_valuation_def],
by_cases hx : x = 0,
{ rw [hx, zero_mul, if_pos (eq.refl _), zero_mul] },
{ by_cases hy : y = 0,
{ rw [hy, mul_zero, if_pos (eq.refl _), mul_zero] },
{ rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← with_zero.coe_mul, with_zero.coe_inj,
← of_add_add,... | lemma | is_dedekind_domain.height_one_spectrum.int_valuation.map_mul' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"associates.count_mul",
"associates.mk_mul_mk",
"ideal.span_singleton_mul_span_singleton",
"mul_ne_zero",
"mul_zero",
"of_add_add",
"with_zero.coe_mul",
"zero_mul"
] | The `v`-adic valuation of a product equals the product of the valuations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation.le_max_iff_min_le {a b c : ℕ} : multiplicative.of_add(-c : ℤ) ≤
max (multiplicative.of_add(-a : ℤ)) (multiplicative.of_add(-b : ℤ)) ↔ min a b ≤ c | by rw [le_max_iff, of_add_le, of_add_le, neg_le_neg_iff, neg_le_neg_iff, int.coe_nat_le,
int.coe_nat_le, ← min_le_iff] | lemma | is_dedekind_domain.height_one_spectrum.int_valuation.le_max_iff_min_le | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"int.coe_nat_le",
"le_max_iff",
"min_le_iff",
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_valuation.map_add_le_max' (x y : R) : v.int_valuation_def (x + y) ≤
max (v.int_valuation_def x) (v.int_valuation_def y) | begin
by_cases hx : x = 0,
{ rw [hx, zero_add],
conv_rhs {rw [int_valuation_def, if_pos (eq.refl _)]},
rw max_eq_right (with_zero.zero_le (v.int_valuation_def y)),
exact le_refl _, },
{ by_cases hy : y = 0,
{ rw [hy, add_zero],
conv_rhs {rw [max_comm, int_valuation_def, if_pos (eq.refl _)]},... | lemma | is_dedekind_domain.height_one_spectrum.int_valuation.map_add_le_max' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"associates.le_singleton_iff",
"associates.mk",
"associates.prime_pow_dvd_iff_le",
"ideal.add_mem",
"ideal.span",
"with_zero.le_max_iff",
"with_zero.zero_le",
"zero_le'"
] | The `v`-adic valuation of a sum is bounded above by the maximum of the valuations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation : valuation R ℤₘ₀ | { to_fun := v.int_valuation_def,
map_zero' := int_valuation.map_zero' v,
map_one' := int_valuation.map_one' v,
map_mul' := int_valuation.map_mul' v,
map_add_le_max' := int_valuation.map_add_le_max' v } | def | is_dedekind_domain.height_one_spectrum.int_valuation | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"valuation"
] | The `v`-adic valuation on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation_exists_uniformizer :
∃ (π : R), v.int_valuation_def π = multiplicative.of_add (-1 : ℤ) | begin
have hv : _root_.irreducible (associates.mk v.as_ideal) := v.associates_irreducible,
have hlt : v.as_ideal^2 < v.as_ideal,
{ rw ← ideal.dvd_not_unit_iff_lt,
exact ⟨v.ne_bot, v.as_ideal,
(not_congr ideal.is_unit_iff).mpr (ideal.is_prime.ne_top v.is_prime), sq v.as_ideal⟩ } ,
obtain ⟨π, mem, nmem⟩ ... | lemma | is_dedekind_domain.height_one_spectrum.int_valuation_exists_uniformizer | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"associates.mk",
"associates.mk_le_mk_iff_dvd_iff",
"associates.mk_ne_zero'",
"associates.mk_pow",
"associates.prime_pow_dvd_iff_le",
"ideal.dvd_not_unit_iff_lt",
"ideal.dvd_span_singleton",
"ideal.is_prime.ne_top",
"ideal.is_unit_iff",
"ideal.span",
"int.coe_nat_inj'",
"multiplicative.of_add"... | There exists `π ∈ R` with `v`-adic valuation `multiplicative.of_add (-1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation (v : height_one_spectrum R) : valuation K ℤₘ₀ | v.int_valuation.extend_to_localization (λ r hr, set.mem_compl $ v.int_valuation_ne_zero' ⟨r, hr⟩) K | def | is_dedekind_domain.height_one_spectrum.valuation | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"set.mem_compl",
"valuation"
] | The `v`-adic valuation of `x ∈ K` is the valuation of `r` divided by the valuation of `s`,
where `r` and `s` are chosen so that `x = r/s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_def (x : K) : v.valuation x = v.int_valuation.extend_to_localization
(λ r hr, set.mem_compl (v.int_valuation_ne_zero' ⟨r, hr⟩)) K x | rfl | lemma | is_dedekind_domain.height_one_spectrum.valuation_def | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"set.mem_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_of_mk' {r : R} {s : non_zero_divisors R} :
v.valuation (is_localization.mk' K r s) = v.int_valuation r / v.int_valuation s | begin
erw [valuation_def, (is_localization.to_localization_map (non_zero_divisors R) K).lift_mk',
div_eq_mul_inv, mul_eq_mul_left_iff],
left,
rw [units.coe_inv, inv_inj],
refl,
end | lemma | is_dedekind_domain.height_one_spectrum.valuation_of_mk' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"div_eq_mul_inv",
"inv_inj",
"is_localization.mk'",
"is_localization.to_localization_map",
"mul_eq_mul_left_iff",
"non_zero_divisors",
"units.coe_inv"
] | The `v`-adic valuation of `r/s ∈ K` is the valuation of `r` divided by the valuation of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_of_algebra_map (r : R) :
v.valuation (algebra_map R K r) = v.int_valuation r | by rw [valuation_def, valuation.extend_to_localization_apply_map_apply] | lemma | is_dedekind_domain.height_one_spectrum.valuation_of_algebra_map | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"algebra_map",
"valuation.extend_to_localization_apply_map_apply"
] | The `v`-adic valuation on `K` extends the `v`-adic valuation on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_le_one (r : R) : v.valuation (algebra_map R K r) ≤ 1 | by { rw valuation_of_algebra_map, exact v.int_valuation_le_one r } | lemma | is_dedekind_domain.height_one_spectrum.valuation_le_one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"algebra_map"
] | The `v`-adic valuation on `R` is bounded above by 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_lt_one_iff_dvd (r : R) :
v.valuation (algebra_map R K r) < 1 ↔ v.as_ideal ∣ ideal.span {r} | by { rw valuation_of_algebra_map, exact v.int_valuation_lt_one_iff_dvd r } | lemma | is_dedekind_domain.height_one_spectrum.valuation_lt_one_iff_dvd | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"algebra_map",
"ideal.span"
] | The `v`-adic valuation of `r ∈ R` is less than 1 if and only if `v` divides the ideal `(r)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_exists_uniformizer :
∃ (π : K), v.valuation π = multiplicative.of_add (-1 : ℤ) | begin
obtain ⟨r, hr⟩ := v.int_valuation_exists_uniformizer,
use algebra_map R K r,
rw [valuation_def, valuation.extend_to_localization_apply_map_apply],
exact hr,
end | lemma | is_dedekind_domain.height_one_spectrum.valuation_exists_uniformizer | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"algebra_map",
"multiplicative.of_add",
"valuation.extend_to_localization_apply_map_apply"
] | There exists `π ∈ K` with `v`-adic valuation `multiplicative.of_add (-1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_uniformizer_ne_zero :
(classical.some (v.valuation_exists_uniformizer K)) ≠ 0 | begin
have hu := classical.some_spec (v.valuation_exists_uniformizer K),
exact (valuation.ne_zero_iff _).mp (ne_of_eq_of_ne hu with_zero.coe_ne_zero),
end | lemma | is_dedekind_domain.height_one_spectrum.valuation_uniformizer_ne_zero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"valuation.ne_zero_iff"
] | Uniformizers are nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adic_valued : valued K ℤₘ₀ | valued.mk' v.valuation | def | is_dedekind_domain.height_one_spectrum.adic_valued | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"valued",
"valued.mk'"
] | `K` as a valued field with the `v`-adic valuation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adic_valued_apply {x : K} : (v.adic_valued.v : _) x = v.valuation x | rfl | lemma | is_dedekind_domain.height_one_spectrum.adic_valued_apply | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_completion | @uniform_space.completion K v.adic_valued.to_uniform_space | def | is_dedekind_domain.height_one_spectrum.adic_completion | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"adic_completion",
"uniform_space.completion"
] | The completion of `K` with respect to its `v`-adic valuation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valued_adic_completion : valued (v.adic_completion K) ℤₘ₀ | @valued.valued_completion _ _ _ _ v.adic_valued | instance | is_dedekind_domain.height_one_spectrum.valued_adic_completion | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"valued",
"valued.valued_completion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valued_adic_completion_def {x : v.adic_completion K} :
valued.v x = @valued.extension K _ _ _ (adic_valued v) x | rfl | lemma | is_dedekind_domain.height_one_spectrum.valued_adic_completion_def | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"valued.extension"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_completion_complete_space : complete_space (v.adic_completion K) | @uniform_space.completion.complete_space K v.adic_valued.to_uniform_space | instance | is_dedekind_domain.height_one_spectrum.adic_completion_complete_space | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"complete_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_completion.has_lift_t : has_lift_t K (v.adic_completion K) | (infer_instance : has_lift_t K (@uniform_space.completion K v.adic_valued.to_uniform_space)) | instance | is_dedekind_domain.height_one_spectrum.adic_completion.has_lift_t | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"uniform_space.completion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_completion_integers : valuation_subring (v.adic_completion K) | valued.v.valuation_subring | def | is_dedekind_domain.height_one_spectrum.adic_completion_integers | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"valuation_subring"
] | The ring of integers of `adic_completion`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_adic_completion_integers {x : v.adic_completion K} :
x ∈ v.adic_completion_integers K ↔ (valued.v x : ℤₘ₀) ≤ 1 | iff.rfl | lemma | is_dedekind_domain.height_one_spectrum.mem_adic_completion_integers | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_valued.has_uniform_continuous_const_smul' :
@has_uniform_continuous_const_smul R K v.adic_valued.to_uniform_space _ | @has_uniform_continuous_const_smul_of_continuous_const_smul R K _ _ _
v.adic_valued.to_uniform_space _ _ | instance | is_dedekind_domain.height_one_spectrum.adic_valued.has_uniform_continuous_const_smul' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"has_uniform_continuous_const_smul",
"has_uniform_continuous_const_smul_of_continuous_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_valued.has_uniform_continuous_const_smul :
@has_uniform_continuous_const_smul K K v.adic_valued.to_uniform_space _ | @ring.has_uniform_continuous_const_smul K _ v.adic_valued.to_uniform_space _ _ | instance | is_dedekind_domain.height_one_spectrum.adic_valued.has_uniform_continuous_const_smul | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"has_uniform_continuous_const_smul",
"ring.has_uniform_continuous_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_completion.algebra' : algebra R (v.adic_completion K) | @uniform_space.completion.algebra K _ v.adic_valued.to_uniform_space _ _ R _ _
(adic_valued.has_uniform_continuous_const_smul' R K v) | instance | is_dedekind_domain.height_one_spectrum.adic_completion.algebra' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul_adic_completion (r : R) (x : K) :
(↑(r • x) : v.adic_completion K) = r • (↑x : v.adic_completion K) | @uniform_space.completion.coe_smul R K v.adic_valued.to_uniform_space _ _ r x | lemma | is_dedekind_domain.height_one_spectrum.coe_smul_adic_completion | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"uniform_space.completion.coe_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_adic_completion' :
⇑(algebra_map R $ v.adic_completion K) = coe ∘ algebra_map R K | rfl | lemma | is_dedekind_domain.height_one_spectrum.algebra_map_adic_completion' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_adic_completion :
⇑(algebra_map K $ v.adic_completion K) = coe | rfl | lemma | is_dedekind_domain.height_one_spectrum.algebra_map_adic_completion | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul_adic_completion_integers (r : R) (x : v.adic_completion_integers K) :
(↑(r • x) : v.adic_completion K) = r • (x : v.adic_completion K) | rfl | lemma | is_dedekind_domain.height_one_spectrum.coe_smul_adic_completion_integers | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adic_completion.is_scalar_tower' :
is_scalar_tower R (v.adic_completion_integers K) (v.adic_completion K) | { smul_assoc := λ x y z, by {simp only [algebra.smul_def], apply mul_assoc, }} | instance | is_dedekind_domain.height_one_spectrum.adic_completion.is_scalar_tower' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"algebra.smul_def",
"is_scalar_tower",
"mul_assoc",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring.dimension_le_one : Prop | ∀ p ≠ (⊥ : ideal R), p.is_prime → p.is_maximal | def | ring.dimension_le_one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/basic.lean | [
"ring_theory.ideal.over",
"ring_theory.polynomial.rational_root"
] | [
"ideal"
] | A ring `R` has Krull dimension at most one if all nonzero prime ideals are maximal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dimension_le_one.principal_ideal_ring
[is_domain A] [is_principal_ideal_ring A] : dimension_le_one A | λ p nonzero prime, by { haveI := prime, exact is_prime.to_maximal_ideal nonzero } | lemma | ring.dimension_le_one.principal_ideal_ring | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/basic.lean | [
"ring_theory.ideal.over",
"ring_theory.polynomial.rational_root"
] | [
"is_domain",
"is_prime.to_maximal_ideal",
"is_principal_ideal_ring",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimension_le_one.is_integral_closure (B : Type*) [comm_ring B] [is_domain B]
[nontrivial R] [algebra R A] [algebra R B] [algebra B A] [is_scalar_tower R B A]
[is_integral_closure B R A] (h : dimension_le_one R) :
dimension_le_one B | λ p ne_bot prime, by exactI
is_integral_closure.is_maximal_of_is_maximal_comap A p
(h _ (is_integral_closure.comap_ne_bot A ne_bot) infer_instance) | lemma | ring.dimension_le_one.is_integral_closure | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/basic.lean | [
"ring_theory.ideal.over",
"ring_theory.polynomial.rational_root"
] | [
"algebra",
"comm_ring",
"is_domain",
"is_integral_closure",
"is_scalar_tower",
"nontrivial",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimension_le_one.integral_closure [nontrivial R] [is_domain A] [algebra R A]
(h : dimension_le_one R) : dimension_le_one (integral_closure R A) | h.is_integral_closure R A (integral_closure R A) | lemma | ring.dimension_le_one.integral_closure | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/basic.lean | [
"ring_theory.ideal.over",
"ring_theory.polynomial.rational_root"
] | [
"algebra",
"integral_closure",
"is_domain",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimension_le_one.not_lt_lt (h : ring.dimension_le_one R)
(p₀ p₁ p₂ : ideal R) [hp₁ : p₁.is_prime] [hp₂ : p₂.is_prime] :
¬ (p₀ < p₁ ∧ p₁ < p₂) | | ⟨h01, h12⟩ := h12.ne ((h p₁ (bot_le.trans_lt h01).ne' hp₁).eq_of_le hp₂.ne_top h12.le) | lemma | ring.dimension_le_one.not_lt_lt | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/basic.lean | [
"ring_theory.ideal.over",
"ring_theory.polynomial.rational_root"
] | [
"ideal",
"ring.dimension_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dimension_le_one.eq_bot_of_lt (h : ring.dimension_le_one R)
(p P : ideal R) [hp : p.is_prime] [hP : P.is_prime] (hpP : p < P) : p = ⊥ | by_contra (λ hp0, h.not_lt_lt ⊥ p P ⟨ne.bot_lt hp0, hpP⟩) | lemma | ring.dimension_le_one.eq_bot_of_lt | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/basic.lean | [
"ring_theory.ideal.over",
"ring_theory.polynomial.rational_root"
] | [
"by_contra",
"ideal",
"ring.dimension_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_dedekind_domain : Prop | (is_noetherian_ring : is_noetherian_ring A)
(dimension_le_one : dimension_le_one A)
(is_integrally_closed : is_integrally_closed A) | class | is_dedekind_domain | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/basic.lean | [
"ring_theory.ideal.over",
"ring_theory.polynomial.rational_root"
] | [
"is_integrally_closed",
"is_noetherian_ring"
] | A Dedekind domain is an integral domain that is Noetherian, integrally closed, and
has Krull dimension at most one.
This is definition 3.2 of [Neukirch1992].
The integral closure condition is independent of the choice of field of fractions:
use `is_dedekind_domain_iff` to prove `is_dedekind_domain` for a given `fract... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain_iff (K : Type*) [field K] [algebra A K] [is_fraction_ring A K] :
is_dedekind_domain A ↔ is_noetherian_ring A ∧ dimension_le_one A ∧
(∀ {x : K}, is_integral A x → ∃ y, algebra_map A K y = x) | ⟨λ ⟨hr, hd, hi⟩, ⟨hr, hd, λ x, (is_integrally_closed_iff K).mp hi⟩,
λ ⟨hr, hd, hi⟩, ⟨hr, hd, (is_integrally_closed_iff K).mpr @hi⟩⟩ | lemma | is_dedekind_domain_iff | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/basic.lean | [
"ring_theory.ideal.over",
"ring_theory.polynomial.rational_root"
] | [
"algebra",
"algebra_map",
"field",
"is_dedekind_domain",
"is_fraction_ring",
"is_integral",
"is_integrally_closed_iff",
"is_noetherian_ring"
] | An integral domain is a Dedekind domain iff and only if it is
Noetherian, has dimension ≤ 1, and is integrally closed in a given fraction field.
In particular, this definition does not depend on the choice of this fraction field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_principal_ideal_ring.is_dedekind_domain [is_principal_ideal_ring A] :
is_dedekind_domain A | ⟨principal_ideal_ring.is_noetherian_ring,
ring.dimension_le_one.principal_ideal_ring A,
unique_factorization_monoid.is_integrally_closed⟩ | instance | is_principal_ideal_ring.is_dedekind_domain | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/basic.lean | [
"ring_theory.ideal.over",
"ring_theory.polynomial.rational_root"
] | [
"is_dedekind_domain",
"is_principal_ideal_ring",
"ring.dimension_le_one.principal_ideal_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_dedekind_domain_dvr : Prop | (is_noetherian_ring : is_noetherian_ring A)
(is_dvr_at_nonzero_prime : ∀ P ≠ (⊥ : ideal A), P.is_prime →
discrete_valuation_ring (localization.at_prime P)) | structure | is_dedekind_domain_dvr | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/dvr.lean | [
"ring_theory.localization.localization_localization",
"ring_theory.localization.submodule",
"ring_theory.discrete_valuation_ring.tfae"
] | [
"discrete_valuation_ring",
"ideal",
"is_noetherian_ring",
"localization.at_prime"
] | A Dedekind domain is an integral domain that is Noetherian, and the
localization at every nonzero prime is a discrete valuation ring.
This is equivalent to `is_dedekind_domain`.
TODO: prove the equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring.dimension_le_one.localization {R : Type*} (Rₘ : Type*) [comm_ring R] [is_domain R]
[comm_ring Rₘ] [algebra R Rₘ] {M : submonoid R} [is_localization M Rₘ] (hM : M ≤ R⁰)
(h : ring.dimension_le_one R) : ring.dimension_le_one Rₘ | begin
introsI p hp0 hpp,
refine ideal.is_maximal_def.mpr ⟨hpp.ne_top, ideal.maximal_of_no_maximal (λ P hpP hPm, _)⟩,
have hpP' : (⟨p, hpp⟩ : {p : ideal Rₘ // p.is_prime}) < ⟨P, hPm.is_prime⟩ := hpP,
rw ← (is_localization.order_iso_of_prime M Rₘ).lt_iff_lt at hpP',
haveI : ideal.is_prime (ideal.comap (algebra_... | lemma | ring.dimension_le_one.localization | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/dvr.lean | [
"ring_theory.localization.localization_localization",
"ring_theory.localization.submodule",
"ring_theory.discrete_valuation_ring.tfae"
] | [
"algebra",
"algebra_map",
"comm_ring",
"ideal",
"ideal.comap",
"ideal.is_prime",
"ideal.maximal_of_no_maximal",
"is_domain",
"is_localization",
"is_localization.bot_lt_comap_prime",
"is_localization.order_iso_of_prime",
"ring.dimension_le_one",
"submonoid"
] | Localizing a domain of Krull dimension `≤ 1` gives another ring of Krull dimension `≤ 1`.
Note that the same proof can/should be generalized to preserving any Krull dimension,
once we have a suitable definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization.is_dedekind_domain [is_dedekind_domain A] {M : submonoid A} (hM : M ≤ A⁰)
(Aₘ : Type*) [comm_ring Aₘ] [is_domain Aₘ] [algebra A Aₘ]
[is_localization M Aₘ] : is_dedekind_domain Aₘ | begin
have : ∀ (y : M), is_unit (algebra_map A (fraction_ring A) y),
{ rintros ⟨y, hy⟩,
exact is_unit.mk0 _ (mt is_fraction_ring.to_map_eq_zero_iff.mp (non_zero_divisors.ne_zero
(hM hy))) },
letI : algebra Aₘ (fraction_ring A) := ring_hom.to_algebra (is_localization.lift this),
haveI : is_scalar_tower... | lemma | is_localization.is_dedekind_domain | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/dvr.lean | [
"ring_theory.localization.localization_localization",
"ring_theory.localization.submodule",
"ring_theory.discrete_valuation_ring.tfae"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"comm_ring",
"fraction_ring",
"is_dedekind_domain",
"is_dedekind_domain_iff",
"is_domain",
"is_fraction_ring",
"is_fraction_ring.is_fraction_ring_of_is_domain_of_is_localization",
"is_integrally_closed_iff",
"is_localization",
"is_localization.is... | The localization of a Dedekind domain is a Dedekind domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization.at_prime.is_dedekind_domain [is_dedekind_domain A]
(P : ideal A) [P.is_prime] (Aₘ : Type*) [comm_ring Aₘ] [is_domain Aₘ] [algebra A Aₘ]
[is_localization.at_prime Aₘ P] : is_dedekind_domain Aₘ | is_localization.is_dedekind_domain A P.prime_compl_le_non_zero_divisors Aₘ | lemma | is_localization.at_prime.is_dedekind_domain | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/dvr.lean | [
"ring_theory.localization.localization_localization",
"ring_theory.localization.submodule",
"ring_theory.discrete_valuation_ring.tfae"
] | [
"algebra",
"comm_ring",
"ideal",
"is_dedekind_domain",
"is_domain",
"is_localization.at_prime",
"is_localization.is_dedekind_domain"
] | The localization of a Dedekind domain at every nonzero prime ideal is a Dedekind domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization.at_prime.not_is_field
{P : ideal A} (hP : P ≠ ⊥) [pP : P.is_prime]
(Aₘ : Type*) [comm_ring Aₘ] [algebra A Aₘ] [is_localization.at_prime Aₘ P] :
¬ (is_field Aₘ) | begin
intro h,
letI := h.to_field,
obtain ⟨x, x_mem, x_ne⟩ := P.ne_bot_iff.mp hP,
exact (local_ring.maximal_ideal.is_maximal _).ne_top (ideal.eq_top_of_is_unit_mem _
((is_localization.at_prime.to_map_mem_maximal_iff Aₘ P _).mpr x_mem)
(is_unit_iff_ne_zero.mpr ((map_ne_zero_iff (algebra_map A Aₘ)
(... | lemma | is_localization.at_prime.not_is_field | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/dvr.lean | [
"ring_theory.localization.localization_localization",
"ring_theory.localization.submodule",
"ring_theory.discrete_valuation_ring.tfae"
] | [
"algebra",
"algebra_map",
"comm_ring",
"ideal",
"ideal.eq_top_of_is_unit_mem",
"is_field",
"is_localization.at_prime",
"is_localization.at_prime.to_map_mem_maximal_iff",
"is_localization.injective",
"local_ring.maximal_ideal.is_maximal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.at_prime.discrete_valuation_ring_of_dedekind_domain [is_dedekind_domain A]
{P : ideal A} (hP : P ≠ ⊥) [pP : P.is_prime]
(Aₘ : Type*) [comm_ring Aₘ] [is_domain Aₘ] [algebra A Aₘ] [is_localization.at_prime Aₘ P] :
discrete_valuation_ring Aₘ | begin
classical,
letI : is_noetherian_ring Aₘ := is_localization.is_noetherian_ring P.prime_compl _
is_dedekind_domain.is_noetherian_ring,
letI : local_ring Aₘ := is_localization.at_prime.local_ring Aₘ P,
have hnf := is_localization.at_prime.not_is_field A hP Aₘ,
exact ((discrete_valuation_ring.tfae Aₘ hn... | lemma | is_localization.at_prime.discrete_valuation_ring_of_dedekind_domain | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/dvr.lean | [
"ring_theory.localization.localization_localization",
"ring_theory.localization.submodule",
"ring_theory.discrete_valuation_ring.tfae"
] | [
"algebra",
"comm_ring",
"discrete_valuation_ring",
"discrete_valuation_ring.tfae",
"ideal",
"is_dedekind_domain",
"is_domain",
"is_localization.at_prime",
"is_localization.at_prime.is_dedekind_domain",
"is_localization.at_prime.local_ring",
"is_localization.at_prime.not_is_field",
"is_localiza... | In a Dedekind domain, the localization at every nonzero prime ideal is a DVR. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain.is_dedekind_domain_dvr [is_dedekind_domain A] :
is_dedekind_domain_dvr A | { is_noetherian_ring := is_dedekind_domain.is_noetherian_ring,
is_dvr_at_nonzero_prime := λ P hP pP, by exactI
is_localization.at_prime.discrete_valuation_ring_of_dedekind_domain A hP _ } | theorem | is_dedekind_domain.is_dedekind_domain_dvr | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/dvr.lean | [
"ring_theory.localization.localization_localization",
"ring_theory.localization.submodule",
"ring_theory.discrete_valuation_ring.tfae"
] | [
"is_dedekind_domain",
"is_dedekind_domain_dvr",
"is_localization.at_prime.discrete_valuation_ring_of_dedekind_domain",
"is_noetherian_ring"
] | Dedekind domains, in the sense of Noetherian integrally closed domains of Krull dimension ≤ 1,
are also Dedekind domains in the sense of Noetherian domains where the localization at every
nonzero prime ideal is a DVR. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_dedekind_domain.height_one_spectrum.max_pow_dividing (I : ideal R) : ideal R | v.as_ideal^(associates.mk v.as_ideal).count (associates.mk I).factors | def | is_dedekind_domain.height_one_spectrum.max_pow_dividing | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associates.mk",
"ideal"
] | Given a maximal ideal `v` and an ideal `I` of `R`, `max_pow_dividing` returns the maximal
power of `v` dividing `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.finite_factors {I : ideal R} (hI : I ≠ 0) :
{v : height_one_spectrum R | v.as_ideal ∣ I}.finite | begin
rw [← set.finite_coe_iff, set.coe_set_of],
haveI h_fin := fintype_subtype_dvd I hI,
refine finite.of_injective (λ v, (⟨(v : height_one_spectrum R).as_ideal, v.2⟩ : {x // x ∣ I})) _,
intros v w hvw,
simp only at hvw,
exact subtype.coe_injective ((height_one_spectrum.ext_iff ↑v ↑w).mpr hvw)
end | lemma | ideal.finite_factors | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"finite",
"finite.of_injective",
"ideal",
"set.coe_set_of",
"set.finite_coe_iff",
"subtype.coe_injective"
] | Only finitely many maximal ideals of `R` divide a given nonzero ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associates.finite_factors {I : ideal R} (hI : I ≠ 0) :
∀ᶠ (v : height_one_spectrum R) in filter.cofinite,
((associates.mk v.as_ideal).count (associates.mk I).factors : ℤ) = 0 | begin
have h_supp : {v : height_one_spectrum R |
¬((associates.mk v.as_ideal).count (associates.mk I).factors : ℤ) = 0} =
{v : height_one_spectrum R | v.as_ideal ∣ I},
{ ext v,
simp_rw int.coe_nat_eq_zero,
exact associates.count_ne_zero_iff_dvd hI v.irreducible, },
rw [filter.eventually_cofinite, ... | lemma | associates.finite_factors | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associates.count_ne_zero_iff_dvd",
"associates.mk",
"filter.cofinite",
"filter.eventually_cofinite",
"ideal",
"ideal.finite_factors",
"int.coe_nat_eq_zero"
] | For every nonzero ideal `I` of `v`, there are finitely many maximal ideals `v` such that the
multiplicity of `v` in the factorization of `I`, denoted `val_v(I)`, is nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_mul_support {I : ideal R} (hI : I ≠ 0) :
(mul_support (λ (v : height_one_spectrum R), v.max_pow_dividing I)).finite | begin
have h_subset : {v : height_one_spectrum R | v.max_pow_dividing I ≠ 1} ⊆
{v : height_one_spectrum R |
((associates.mk v.as_ideal).count (associates.mk I).factors : ℤ) ≠ 0},
{ intros v hv h_zero,
have hv' : v.max_pow_dividing I = 1,
{ rw [is_dedekind_domain.height_one_spectrum.max_pow_dividin... | lemma | ideal.finite_mul_support | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associates.finite_factors",
"associates.mk",
"finite",
"ideal",
"is_dedekind_domain.height_one_spectrum.max_pow_dividing",
"pow_zero"
] | For every nonzero ideal `I` of `v`, there are finitely many maximal ideals `v` such that
`v^(val_v(I))` is not the unit ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_mul_support_coe {I : ideal R} (hI : I ≠ 0) :
(mul_support (λ (v : height_one_spectrum R),
(v.as_ideal : fractional_ideal R⁰ K) ^
((associates.mk v.as_ideal).count (associates.mk I).factors : ℤ))).finite | begin
rw mul_support,
simp_rw [ne.def, zpow_coe_nat, ← fractional_ideal.coe_ideal_pow,
fractional_ideal.coe_ideal_eq_one],
exact finite_mul_support hI,
end | lemma | ideal.finite_mul_support_coe | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associates.mk",
"finite",
"fractional_ideal",
"fractional_ideal.coe_ideal_eq_one",
"fractional_ideal.coe_ideal_pow",
"ideal",
"zpow_coe_nat"
] | For every nonzero ideal `I` of `v`, there are finitely many maximal ideals `v` such that
`v^(val_v(I))`, regarded as a fractional ideal, is not `(1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_mul_support_inv {I : ideal R} (hI : I ≠ 0) :
(mul_support (λ (v : height_one_spectrum R),
(v.as_ideal : fractional_ideal R⁰ K) ^
-((associates.mk v.as_ideal).count (associates.mk I).factors : ℤ))).finite | begin
rw mul_support,
simp_rw [zpow_neg, ne.def, inv_eq_one],
exact finite_mul_support_coe hI,
end | lemma | ideal.finite_mul_support_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associates.mk",
"finite",
"fractional_ideal",
"ideal",
"inv_eq_one",
"zpow_neg"
] | For every nonzero ideal `I` of `v`, there are finitely many maximal ideals `v` such that
`v^-(val_v(I))` is not the unit ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_not_dvd (I : ideal R) (hI : I ≠ 0) :
¬ (v.as_ideal) ^ ((associates.mk v.as_ideal).count (associates.mk I).factors + 1) ∣
(∏ᶠ (v : height_one_spectrum R), v.max_pow_dividing I) | begin
have hf := finite_mul_support hI,
have h_ne_zero : v.max_pow_dividing I ≠ 0 := pow_ne_zero _ v.ne_bot,
rw [← mul_finprod_cond_ne v hf, pow_add, pow_one, finprod_cond_ne _ _ hf],
intro h_contr,
have hv_prime : prime v.as_ideal := ideal.prime_of_is_prime v.ne_bot v.is_prime,
obtain ⟨w, hw, hvw'⟩ :=
... | lemma | ideal.finprod_not_dvd | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associated_iff_eq",
"associates.mk",
"finprod_cond_ne",
"ideal",
"ideal.prime_of_is_prime",
"mul_dvd_mul_iff_left",
"mul_finprod_cond_ne",
"pow_add",
"pow_ne_zero",
"pow_one",
"prime",
"prime.dvd_of_dvd_pow",
"prime.dvd_prime_iff_associated",
"prime.exists_mem_finset_dvd"
] | For every nonzero ideal `I` of `v`, `v^(val_v(I) + 1)` does not divide `∏_v v^(val_v(I))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
associates.finprod_ne_zero (I : ideal R) :
associates.mk (∏ᶠ (v : height_one_spectrum R), v.max_pow_dividing I) ≠ 0 | begin
rw [associates.mk_ne_zero, finprod_def],
split_ifs,
{ rw finset.prod_ne_zero_iff,
intros v hv,
apply pow_ne_zero _ v.ne_bot, },
{ exact one_ne_zero, }
end | lemma | associates.finprod_ne_zero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associates.mk",
"associates.mk_ne_zero",
"finprod_def",
"finset.prod_ne_zero_iff",
"ideal",
"one_ne_zero",
"pow_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finprod_count (I : ideal R) (hI : I ≠ 0) : (associates.mk v.as_ideal).count
(associates.mk (∏ᶠ (v : height_one_spectrum R), v.max_pow_dividing I)).factors =
(associates.mk v.as_ideal).count (associates.mk I).factors | begin
have h_ne_zero := associates.finprod_ne_zero I,
have hv : irreducible (associates.mk v.as_ideal) := v.associates_irreducible,
have h_dvd := finprod_mem_dvd v (ideal.finite_mul_support hI),
have h_not_dvd := ideal.finprod_not_dvd v I hI,
simp only [is_dedekind_domain.height_one_spectrum.max_pow_dividing]... | lemma | ideal.finprod_count | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associates.dvd_eq_le",
"associates.finprod_ne_zero",
"associates.mk",
"associates.mk_dvd_mk",
"associates.mk_pow",
"associates.prime_pow_dvd_iff_le",
"finprod_mem_dvd",
"ideal",
"ideal.finite_mul_support",
"ideal.finprod_not_dvd",
"irreducible",
"is_dedekind_domain.height_one_spectrum.max_pow... | The multiplicity of `v` in `∏_v v^(val_v(I))` equals `val_v(I)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_height_one_spectrum_factorization (I : ideal R) (hI : I ≠ 0) :
∏ᶠ (v : height_one_spectrum R), v.max_pow_dividing I = I | begin
rw [← associated_iff_eq, ← associates.mk_eq_mk_iff_associated],
apply associates.eq_of_eq_counts,
{ apply associates.finprod_ne_zero I },
{ apply associates.mk_ne_zero.mpr hI },
intros v hv,
obtain ⟨J, hJv⟩ := associates.exists_rep v,
rw [← hJv, associates.irreducible_mk] at hv,
rw ← hJv,
apply ... | lemma | ideal.finprod_height_one_spectrum_factorization | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associated_iff_eq",
"associates.eq_of_eq_counts",
"associates.exists_rep",
"associates.finprod_ne_zero",
"associates.irreducible_mk",
"associates.mk_eq_mk_iff_associated",
"ideal",
"ideal.finprod_count",
"ideal.is_prime_of_prime",
"irreducible.ne_zero"
] | The ideal `I` equals the finprod `∏_v v^(val_v(I))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finprod_height_one_spectrum_factorization_coe (I : ideal R) (hI : I ≠ 0) :
∏ᶠ (v : height_one_spectrum R), (v.as_ideal : fractional_ideal R⁰ K) ^
((associates.mk v.as_ideal).count (associates.mk I).factors : ℤ) = I | begin
conv_rhs { rw ← ideal.finprod_height_one_spectrum_factorization I hI },
rw fractional_ideal.coe_ideal_finprod R⁰ K (le_refl _),
simp_rw [is_dedekind_domain.height_one_spectrum.max_pow_dividing, fractional_ideal.coe_ideal_pow,
zpow_coe_nat],
end | lemma | ideal.finprod_height_one_spectrum_factorization_coe | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/factorization.lean | [
"ring_theory.dedekind_domain.ideal"
] | [
"associates.mk",
"fractional_ideal",
"fractional_ideal.coe_ideal_finprod",
"fractional_ideal.coe_ideal_pow",
"ideal",
"ideal.finprod_height_one_spectrum_factorization",
"is_dedekind_domain.height_one_spectrum.max_pow_dividing",
"zpow_coe_nat"
] | The ideal `I` equals the finprod `∏_v v^(val_v(I))`, when both sides are regarded as fractional
ideals of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_integral_adeles : Type* | Π (v : height_one_spectrum R), v.adic_completion_integers K | def | dedekind_domain.finite_integral_adeles | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [] | The product of all `adic_completion_integers`, where `v` runs over the maximal ideals of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_adic_completions | Π (v : height_one_spectrum R), v.adic_completion K | def | dedekind_domain.prod_adic_completions | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [] | The product of all `adic_completion`, where `v` runs over the maximal ideals of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_apply (x : R_hat R K) (v : height_one_spectrum R) : (x : K_hat R K) v = ↑(x v) | rfl | lemma | dedekind_domain.finite_integral_adeles.coe_apply | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe.add_monoid_hom : add_monoid_hom (R_hat R K) (K_hat R K) | { to_fun := coe,
map_zero' := rfl,
map_add' := λ x y, by { ext v, simp only [coe_apply, pi.add_apply, subring.coe_add] }} | def | dedekind_domain.finite_integral_adeles.coe.add_monoid_hom | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"add_monoid_hom",
"subring.coe_add"
] | The inclusion of `R_hat` in `K_hat` as a homomorphism of additive monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe.ring_hom : ring_hom (R_hat R K) (K_hat R K) | { to_fun := coe,
map_one' := rfl,
map_mul' := λ x y, by {ext p, simp only [pi.mul_apply, subring.coe_mul], refl },
..coe.add_monoid_hom R K } | def | dedekind_domain.finite_integral_adeles.coe.ring_hom | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"pi.mul_apply",
"ring_hom",
"subring.coe_mul"
] | The inclusion of `R_hat` in `K_hat` as a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_adic_completions.algebra' : algebra R (K_hat R K) | (by apply_instance : algebra R $ Π v : height_one_spectrum R, v.adic_completion K) | instance | dedekind_domain.prod_adic_completions.algebra' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_adic_completions.algebra_completions : algebra (R_hat R K) (K_hat R K) | (finite_integral_adeles.coe.ring_hom R K).to_algebra | instance | dedekind_domain.prod_adic_completions.algebra_completions | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_adic_completions.is_scalar_tower_completions :
is_scalar_tower R (R_hat R K) (K_hat R K) | (by apply_instance : is_scalar_tower R (Π v : height_one_spectrum R, v.adic_completion_integers K) $
Π v : height_one_spectrum R, v.adic_completion K) | instance | dedekind_domain.prod_adic_completions.is_scalar_tower_completions | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe.alg_hom : alg_hom R (R_hat R K) (K_hat R K) | { to_fun := coe,
commutes' := λ r, rfl,
..coe.ring_hom R K } | def | dedekind_domain.finite_integral_adeles.coe.alg_hom | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"alg_hom"
] | The inclusion of `R_hat` in `K_hat` as an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe.alg_hom_apply (x : R_hat R K) (v : height_one_spectrum R) :
(coe.alg_hom R K) x v = x v | rfl | lemma | dedekind_domain.finite_integral_adeles.coe.alg_hom_apply | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_adele (x : K_hat R K) | ∀ᶠ v : height_one_spectrum R in filter.cofinite, x v ∈ v.adic_completion_integers K | def | dedekind_domain.prod_adic_completions.is_finite_adele | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"filter.cofinite"
] | An element `x : K_hat R K` is a finite adèle if for all but finitely many height one ideals
`v`, the component `x v` is a `v`-adic integer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add {x y : K_hat R K} (hx : x.is_finite_adele) (hy : y.is_finite_adele) :
(x + y).is_finite_adele | begin
rw [is_finite_adele, filter.eventually_cofinite] at hx hy ⊢,
have h_subset : {v : height_one_spectrum R | ¬ (x + y) v ∈ (v.adic_completion_integers K)} ⊆
{v : height_one_spectrum R | ¬ x v ∈ (v.adic_completion_integers K)} ∪
{v : height_one_spectrum R | ¬ y v ∈ (v.adic_completion_integers K)},
{ in... | lemma | dedekind_domain.prod_adic_completions.is_finite_adele.add | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"filter.eventually_cofinite",
"max_le_iff"
] | The sum of two finite adèles is a finite adèle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero : (0 : K_hat R K).is_finite_adele | begin
rw [is_finite_adele, filter.eventually_cofinite],
have h_empty : {v : height_one_spectrum R |
¬ ((0 : v.adic_completion K) ∈ v.adic_completion_integers K)} = ∅,
{ ext v, rw [mem_empty_iff_false, iff_false], intro hv,
rw mem_set_of_eq at hv, apply hv, rw mem_adic_completion_integers,
have h_zero ... | lemma | dedekind_domain.prod_adic_completions.is_finite_adele.zero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"filter.eventually_cofinite",
"multiplicative",
"zero_le_one'"
] | The tuple `(0)_v` is a finite adèle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg {x : K_hat R K} (hx : x.is_finite_adele) : (-x).is_finite_adele | begin
rw is_finite_adele at hx ⊢,
have h : ∀ (v : height_one_spectrum R), (-x v ∈ v.adic_completion_integers K) ↔
(x v ∈ v.adic_completion_integers K),
{ intro v,
rw [mem_adic_completion_integers, mem_adic_completion_integers, valuation.map_neg], },
simpa only [pi.neg_apply, h] using hx,
end | lemma | dedekind_domain.prod_adic_completions.is_finite_adele.neg | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"valuation.map_neg"
] | The negative of a finite adèle is a finite adèle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul {x y : K_hat R K} (hx : x.is_finite_adele) (hy : y.is_finite_adele) :
(x * y).is_finite_adele | begin
rw [is_finite_adele, filter.eventually_cofinite] at hx hy ⊢,
have h_subset : {v : height_one_spectrum R | ¬ (x * y) v ∈ (v.adic_completion_integers K)} ⊆
{v : height_one_spectrum R | ¬ x v ∈ (v.adic_completion_integers K)} ∪
{v : height_one_spectrum R | ¬ y v ∈ (v.adic_completion_integers K)},
{ in... | lemma | dedekind_domain.prod_adic_completions.is_finite_adele.mul | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"filter.eventually_cofinite",
"multiplicative",
"ordered_comm_monoid.to_covariant_class_left",
"pi.mul_apply"
] | The product of two finite adèles is a finite adèle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one : (1 : K_hat R K).is_finite_adele | begin
rw [is_finite_adele, filter.eventually_cofinite],
have h_empty : {v : height_one_spectrum R |
¬ ((1 : v.adic_completion K) ∈ v.adic_completion_integers K)} = ∅,
{ ext v, rw [mem_empty_iff_false, iff_false], intro hv,
rw mem_set_of_eq at hv, apply hv, rw mem_adic_completion_integers,
exact le_of_... | lemma | dedekind_domain.prod_adic_completions.is_finite_adele.one | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [
"filter.eventually_cofinite",
"pi.one_apply"
] | The tuple `(1)_v` is a finite adèle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_finite_adele_ring_iff (x : K_hat R K) :
x ∈ finite_adele_ring R K ↔ x.is_finite_adele | iff.rfl | lemma | dedekind_domain.mem_finite_adele_ring_iff | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/finite_adele_ring.lean | [
"ring_theory.dedekind_domain.adic_valuation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq : I⁻¹ = 1 / I | rfl | lemma | fractional_ideal.inv_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_zero' : (0 : fractional_ideal R₁⁰ K)⁻¹ = 0 | div_zero | lemma | fractional_ideal.inv_zero' | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"div_zero",
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_nonzero {J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :
J⁻¹ = ⟨(1 : fractional_ideal R₁⁰ K) / J, fractional_div_of_nonzero h⟩ | div_nonzero _ | lemma | fractional_ideal.inv_nonzero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv_of_nonzero {J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :
(↑J⁻¹ : submodule R₁ K) = is_localization.coe_submodule K ⊤ / J | by { rwa inv_nonzero _, refl, assumption } | lemma | fractional_ideal.coe_inv_of_nonzero | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"is_localization.coe_submodule",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inv_iff (hI : I ≠ 0) {x : K} : x ∈ I⁻¹ ↔ ∀ y ∈ I, x * y ∈ (1 : fractional_ideal R₁⁰ K) | mem_div_iff_of_nonzero hI | lemma | fractional_ideal.mem_inv_iff | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_anti_mono (hI : I ≠ 0) (hJ : J ≠ 0) (hIJ : I ≤ J) : J⁻¹ ≤ I⁻¹ | λ x, by { simp only [mem_inv_iff hI, mem_inv_iff hJ], exact λ h y hy, h y (hIJ hy) } | lemma | fractional_ideal.inv_anti_mono | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_self_mul_inv {I : fractional_ideal R₁⁰ K} (hI : I ≤ (1 : fractional_ideal R₁⁰ K)) :
I ≤ I * I⁻¹ | le_self_mul_one_div hI | lemma | fractional_ideal.le_self_mul_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_le_self_mul_inv (I : ideal R₁) : (I : fractional_ideal R₁⁰ K) ≤ I * I⁻¹ | le_self_mul_inv coe_ideal_le_one | lemma | fractional_ideal.coe_ideal_le_self_mul_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inverse_eq (I J : fractional_ideal R₁⁰ K) (h : I * J = 1) : J = I⁻¹ | begin
have hI : I ≠ 0 := ne_zero_of_mul_eq_one I J h,
suffices h' : I * (1 / I) = 1,
{ exact (congr_arg units.inv $
@units.ext _ _ (units.mk_of_mul_eq_one _ _ h) (units.mk_of_mul_eq_one _ _ h') rfl) },
apply le_antisymm,
{ apply mul_le.mpr _,
intros x hx y hy,
rw mul_comm,
exact (mem_div_iff... | theorem | fractional_ideal.right_inverse_eq | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"mul_comm",
"units.ext",
"units.mk_of_mul_eq_one"
] | `I⁻¹` is the inverse of `I` if `I` has an inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_cancel_iff {I : fractional_ideal R₁⁰ K} : I * I⁻¹ = 1 ↔ ∃ J, I * J = 1 | ⟨λ h, ⟨I⁻¹, h⟩, λ ⟨J, hJ⟩, by rwa ← right_inverse_eq K I J hJ⟩ | theorem | fractional_ideal.mul_inv_cancel_iff | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_iff_is_unit {I : fractional_ideal R₁⁰ K} : I * I⁻¹ = 1 ↔ is_unit I | (mul_inv_cancel_iff K).trans is_unit_iff_exists_inv.symm | lemma | fractional_ideal.mul_inv_cancel_iff_is_unit | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv (I : fractional_ideal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(I⁻¹).map (h : K →ₐ[R₁] K') = (I.map h)⁻¹ | by rw [inv_eq, map_div, map_one, inv_eq] | lemma | fractional_ideal.map_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"map_div",
"map_inv",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_inv (x : K) : (span_singleton R₁⁰ x)⁻¹ = span_singleton _ x⁻¹ | one_div_span_singleton x | lemma | fractional_ideal.span_singleton_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_div_span_singleton (x y : K) :
span_singleton R₁⁰ x / span_singleton R₁⁰ y = span_singleton R₁⁰ (x / y) | by rw [div_span_singleton, mul_comm, span_singleton_mul_span_singleton, div_eq_mul_inv] | lemma | fractional_ideal.span_singleton_div_span_singleton | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"div_eq_mul_inv",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_div_self {x : K} (hx : x ≠ 0) :
span_singleton R₁⁰ x / span_singleton R₁⁰ x = 1 | by rw [span_singleton_div_span_singleton, div_self hx, span_singleton_one] | lemma | fractional_ideal.span_singleton_div_self | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"div_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_span_singleton_div_self {x : R₁} (hx : x ≠ 0) :
(ideal.span ({x} : set R₁) : fractional_ideal R₁⁰ K) / ideal.span ({x} : set R₁) = 1 | by rw [coe_ideal_span_singleton, span_singleton_div_self K $
(map_ne_zero_iff _ $ no_zero_smul_divisors.algebra_map_injective R₁ K).mpr hx] | lemma | fractional_ideal.coe_ideal_span_singleton_div_self | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"fractional_ideal",
"ideal.span",
"no_zero_smul_divisors.algebra_map_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_inv {x : K} (hx : x ≠ 0) :
span_singleton R₁⁰ x * (span_singleton R₁⁰ x)⁻¹ = 1 | by rw [span_singleton_inv, span_singleton_mul_span_singleton, mul_inv_cancel hx, span_singleton_one] | lemma | fractional_ideal.span_singleton_mul_inv | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/ideal.lean | [
"algebra.algebra.subalgebra.pointwise",
"algebraic_geometry.prime_spectrum.maximal",
"algebraic_geometry.prime_spectrum.noetherian",
"order.hom.basic",
"ring_theory.dedekind_domain.basic",
"ring_theory.fractional_ideal",
"ring_theory.principal_ideal_domain",
"ring_theory.chain_of_divisors"
] | [
"mul_inv_cancel"
] | 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.