Documentation

Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero

Localizations of commutative monoids with zeroes #

theorem Submonoid.LocalizationMap.subsingleton {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) (h : 0 ∈ S) :

If S contains 0 then the localization at S is trivial.

theorem Submonoid.LocalizationMap.subsingleton_iff {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) :

Subsingleton N ↔ 0 ∈ S

theorem Submonoid.LocalizationMap.nontrivial {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) (h : 0 ∉ S) :

theorem Submonoid.LocalizationMap.map_zero {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) :

f 0 = 0

theorem Submonoid.IsLocalizationMap.map_zero {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] {F : Type u_4} [FunLike F M N] [MulHomClass F M N] {f : F} (hf : S.IsLocalizationMap ⇑f) :

f 0 = 0

instance Submonoid.instMonoidWithZeroHomClassLocalizationMap {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] :

MonoidWithZeroHomClass (S.LocalizationMap N) M N

theorem Localization.mk_zero {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} (x : ↥S) :

mk 0 x = 0

@[instance_reducible]

instance Localization.instCommMonoidWithZero {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} :

CommMonoidWithZero (Localization S)

Equations

Localization.instCommMonoidWithZero = { toCommMonoid := OreLocalization.instCommMonoid, toZero := OreLocalization.instZero, zero_mul := ⋯, mul_zero := ⋯ }

theorem Localization.liftOn_zero {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {p : Type u_4} (f : M → ↥S → p) (H : ∀ {a c : M} {b d : ↥S}, (r S) (a, b) (c, d) → f a b = f c d) :

liftOn 0 f H = f 0 1

@[simp]

theorem Submonoid.LocalizationMap.sec_zero_fst {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] {f : S.LocalizationMap N} :

f (f.sec 0).1 = 0

noncomputable def Submonoid.LocalizationMap.lift₀ {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] {P : Type u_3} [CommMonoidWithZero P] (f : S.LocalizationMap N) (g : M →*₀ P) (hg : ∀ (y : ↥S), IsUnit (g ↑y)) :

Given a Localization map f : M →*₀ N for a Submonoid S ⊆ M and a map of CommMonoidWithZeros g : M →*₀ P such that g y is invertible for all y : S, the homomorphism induced from N to P sending z : N to g x * (g y)⁻¹, where (x, y) : M × S are such that z = f x * (f y)⁻¹.

Equations

f.lift₀ g hg = { toFun := (↑(f.lift hg)).toFun, map_zero' := ⋯, map_one' := ⋯, map_mul' := ⋯ }

Instances For

theorem Submonoid.LocalizationMap.lift₀_def {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] {P : Type u_3} [CommMonoidWithZero P] (f : S.LocalizationMap N) (g : M →*₀ P) (hg : ∀ (y : ↥S), IsUnit (g ↑y)) :

⇑(f.lift₀ g hg) = ⇑(f.lift hg)

theorem Submonoid.LocalizationMap.lift₀_apply {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] {P : Type u_3} [CommMonoidWithZero P] (f : S.LocalizationMap N) (g : M →*₀ P) (hg : ∀ (y : ↥S), IsUnit (g ↑y)) (x : N) :

(f.lift₀ g hg) x = g (f.sec x).1 * ↑((IsUnit.liftRight ((↑g).domRestrict S) hg) (f.sec x).2)⁻¹

theorem Submonoid.LocalizationMap.isCancelMulZero {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) [IsCancelMulZero M] :

IsCancelMulZero N

Given a Localization map f : M →*₀ N for a Submonoid S ⊆ M, if M is a cancellative monoid with zero, and all elements of S are regular, then N is a cancellative monoid with zero.

theorem Submonoid.LocalizationMap.map_eq_zero_iff {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) {m : M} :

f m = 0 ↔ ∃ (s : ↥S), ↑s * m = 0

theorem Submonoid.LocalizationMap.mk'_eq_zero_iff {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) (m : M) (s : ↥S) :

f.mk' m s = 0 ↔ ∃ (s : ↥S), ↑s * m = 0

@[simp]

theorem Submonoid.LocalizationMap.mk'_zero {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) (s : ↥S) :

f.mk' 0 s = 0

theorem Submonoid.LocalizationMap.nonZeroDivisors_le_comap {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) :

nonZeroDivisors M ≤ comap f (nonZeroDivisors N)

theorem Submonoid.LocalizationMap.map_nonZeroDivisors_le {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) :

Submonoid.map f (nonZeroDivisors M) ≤ nonZeroDivisors N

theorem Submonoid.LocalizationMap.noZeroDivisors {M : Type u_1} [CommMonoidWithZero M] {S : Submonoid M} {N : Type u_2} [CommMonoidWithZero N] (f : S.LocalizationMap N) [NoZeroDivisors M] :

NoZeroDivisors N