Direct sum #

This file defines the direct sum of abelian groups, indexed by a discrete type.

Notation #

⨁ i, β i is the n-ary direct sum DirectSum. This notation is in the DirectSum locale, accessible after open DirectSum.

References #

https://en.wikipedia.org/wiki/Direct_sum

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def DirectSum (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] :

Type (max w v)

DirectSum ι β is the direct sum of a family of additive commutative monoids β i.

Note: open DirectSum will enable the notation ⨁ i, β i for DirectSum ι β.

Equations

DirectSum ι β = Π₀ (i : ι), β i

Instances For

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instance instInhabitedDirectSum (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] :

Inhabited (DirectSum ι β)

Equations

instInhabitedDirectSum ι β = inferInstanceAs (Inhabited (Π₀ (i : ι), β i))

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instance instAddCommMonoidDirectSum (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] :

AddCommMonoid (DirectSum ι β)

Equations

instAddCommMonoidDirectSum ι β = inferInstanceAs (AddCommMonoid (Π₀ (i : ι), β i))

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instance instDFunLikeDirectSum (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] :

DFunLike (DirectSum ι β) ι fun (i : ι) => β i

Equations

instDFunLikeDirectSum ι β = inferInstanceAs (DFunLike (Π₀ (i : ι), β i) ι fun (i : ι) => β i)

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instance instCoeFunDirectSumForAll (ι : Type v) (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] :

CoeFun (DirectSum ι β) fun (x : DirectSum ι β) => (i : ι) → β i

Equations

instCoeFunDirectSumForAll ι β = inferInstanceAs (CoeFun (Π₀ (i : ι), β i) fun (x : Π₀ (i : ι), β i) => (i : ι) → β i)

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def DirectSum.«term⨁_,_».delab :

Lean.PrettyPrinter.Delaborator.Delab

Pretty printer defined by notation3 command.

Equations

One or more equations did not get rendered due to their size.

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def DirectSum.«term⨁_,_» :

Lean.ParserDescr

⨁ i, f i is notation for DirectSum _ f and equals the direct sum of fun i ↦ f i. Taking the direct sum over multiple arguments is possible, e.g. ⨁ (i) (j), f i j.

Equations

One or more equations did not get rendered due to their size.

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instance DirectSum.instAddCommGroupDirectSumToAddCommMonoid {ι : Type v} (β : ι → Type w) [(i : ι) → AddCommGroup (β i)] :

AddCommGroup (DirectSum ι β)

Equations

DirectSum.instAddCommGroupDirectSumToAddCommMonoid β = inferInstanceAs (AddCommGroup (Π₀ (i : ι), β i))

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@[simp]

theorem DirectSum.sub_apply {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommGroup (β i)] (g₁ : DirectSum ι fun (i : ι) => β i) (g₂ : DirectSum ι fun (i : ι) => β i) (i : ι) :

(g₁ - g₂) i = g₁ i - g₂ i

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@[simp]

theorem DirectSum.zero_apply {ι : Type v} (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (i : ι) :

0 i = 0

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@[simp]

theorem DirectSum.add_apply {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] (g₁ : DirectSum ι fun (i : ι) => β i) (g₂ : DirectSum ι fun (i : ι) => β i) (i : ι) :

(g₁ + g₂) i = g₁ i + g₂ i

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def DirectSum.mk {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (s : Finset ι) :

((i : ↑↑s) → β ↑i) →+ DirectSum ι fun (i : ι) => β i

mk β s x is the element of ⨁ i, β i that is zero outside s and has coefficient x i for i in s.

Equations

DirectSum.mk β s = { toZeroHom := { toFun := DFinsupp.mk s, map_zero' := ⋯ }, map_add' := ⋯ }

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def DirectSum.of {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (i : ι) :

β i →+ DirectSum ι fun (i : ι) => β i

of i is the natural inclusion map from β i to ⨁ i, β i.

Equations

DirectSum.of β i = DFinsupp.singleAddHom β i

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@[simp]

theorem DirectSum.of_eq_same {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (i : ι) (x : β i) :

((DirectSum.of β i) x) i = x

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theorem DirectSum.of_eq_of_ne {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (i : ι) (j : ι) (x : β i) (h : i ≠ j) :

((DirectSum.of β i) x) j = 0

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@[simp]

theorem DirectSum.support_zero {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] :

DFinsupp.support 0 = ∅

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@[simp]

theorem DirectSum.support_of {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (i : ι) (x : β i) (h : x ≠ 0) :

DFinsupp.support ((DirectSum.of β i) x) = {i}

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theorem DirectSum.support_of_subset {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {i : ι} {b : β i} :

DFinsupp.support ((DirectSum.of β i) b) ⊆ {i}

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theorem DirectSum.sum_support_of {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (x : DirectSum ι fun (i : ι) => β i) :

(Finset.sum (DFinsupp.support x) fun (i : ι) => (DirectSum.of β i) (x i)) = x

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theorem DirectSum.mk_injective {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] (s : Finset ι) :

Function.Injective ⇑(DirectSum.mk β s)

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theorem DirectSum.of_injective {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] (i : ι) :

Function.Injective ⇑(DirectSum.of β i)

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theorem DirectSum.induction_on {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {C : (DirectSum ι fun (i : ι) => β i) → Prop} (x : DirectSum ι fun (i : ι) => β i) (H_zero : C 0) (H_basic : ∀ (i : ι) (x : β i), C ((DirectSum.of β i) x)) (H_plus : ∀ (x y : DirectSum ι fun (i : ι) => β i), C x → C y → C (x + y)) :

C x

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theorem DirectSum.addHom_ext {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u_1} [AddMonoid γ] ⦃f : (DirectSum ι fun (i : ι) => β i) →+ γ⦄ ⦃g : (DirectSum ι fun (i : ι) => β i) →+ γ⦄ (H : ∀ (i : ι) (y : β i), f ((DirectSum.of (fun (i : ι) => β i) i) y) = g ((DirectSum.of (fun (i : ι) => β i) i) y)) :

f = g

If two additive homomorphisms from ⨁ i, β i are equal on each of β i y, then they are equal.

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theorem DirectSum.addHom_ext' {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u_1} [AddMonoid γ] ⦃f : (DirectSum ι fun (i : ι) => β i) →+ γ⦄ ⦃g : (DirectSum ι fun (i : ι) => β i) →+ γ⦄ (H : ∀ (i : ι), AddMonoidHom.comp f (DirectSum.of (fun (i : ι) => β i) i) = AddMonoidHom.comp g (DirectSum.of (fun (i : ι) => β i) i)) :

f = g

If two additive homomorphisms from ⨁ i, β i are equal on each of β i y, then they are equal.

See note [partially-applied ext lemmas].

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def DirectSum.toAddMonoid {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (φ : (i : ι) → β i →+ γ) :

(DirectSum ι fun (i : ι) => β i) →+ γ

toAddMonoid φ is the natural homomorphism from ⨁ i, β i to γ induced by a family φ of homomorphisms β i → γ.

Equations

DirectSum.toAddMonoid φ = DFinsupp.liftAddHom φ

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@[simp]

theorem DirectSum.toAddMonoid_of {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (φ : (i : ι) → β i →+ γ) (i : ι) (x : β i) :

(DirectSum.toAddMonoid φ) ((DirectSum.of β i) x) = (φ i) x

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theorem DirectSum.toAddMonoid.unique {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (ψ : (DirectSum ι fun (i : ι) => β i) →+ γ) (f : DirectSum ι fun (i : ι) => β i) :

ψ f = (DirectSum.toAddMonoid fun (i : ι) => AddMonoidHom.comp ψ (DirectSum.of β i)) f

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theorem DirectSum.toAddMonoid_injective {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] :

Function.Injective DirectSum.toAddMonoid

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@[simp]

theorem DirectSum.toAddMonoid_inj {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] {f : (i : ι) → β i →+ γ} {g : (i : ι) → β i →+ γ} :

DirectSum.toAddMonoid f = DirectSum.toAddMonoid g ↔ f = g

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def DirectSum.fromAddMonoid {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] :

(DirectSum ι fun (i : ι) => γ →+ β i) →+ γ →+ DirectSum ι fun (i : ι) => β i

fromAddMonoid φ is the natural homomorphism from γ to ⨁ i, β i induced by a family φ of homomorphisms γ → β i.

Note that this is not an isomorphism. Not every homomorphism γ →+ ⨁ i, β i arises in this way.

Equations

DirectSum.fromAddMonoid = DirectSum.toAddMonoid fun (i : ι) => AddMonoidHom.compHom (DirectSum.of β i)

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@[simp]

theorem DirectSum.fromAddMonoid_of {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (i : ι) (f : γ →+ β i) :

DirectSum.fromAddMonoid ((DirectSum.of (fun (i : ι) => γ →+ β i) i) f) = AddMonoidHom.comp (DirectSum.of β i) f

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theorem DirectSum.fromAddMonoid_of_apply {ι : Type v} [dec_ι : DecidableEq ι] {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {γ : Type u₁} [AddCommMonoid γ] (i : ι) (f : γ →+ β i) (x : γ) :

(DirectSum.fromAddMonoid ((DirectSum.of (fun (i : ι) => γ →+ β i) i) f)) x = (DirectSum.of β i) (f x)

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def DirectSum.setToSet {ι : Type v} [dec_ι : DecidableEq ι] (β : ι → Type w) [(i : ι) → AddCommMonoid (β i)] (S : Set ι) (T : Set ι) (H : S ⊆ T) :

(DirectSum ↑S fun (i : ↑S) => β ↑i) →+ DirectSum ↑T fun (i : ↑T) => β ↑i

setToSet β S T h is the natural homomorphism ⨁ (i : S), β i → ⨁ (i : T), β i, where h : S ⊆ T.

Equations

DirectSum.setToSet β S T H = DirectSum.toAddMonoid fun (i : ↑S) => DirectSum.of (fun (i : Subtype T) => β ↑i) { val := ↑i, property := ⋯ }

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instance DirectSum.unique {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] [∀ (i : ι), Subsingleton (β i)] :

Unique (DirectSum ι fun (i : ι) => β i)

Equations

DirectSum.unique = DFinsupp.unique

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instance DirectSum.uniqueOfIsEmpty {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] [IsEmpty ι] :

Unique (DirectSum ι fun (i : ι) => β i)

A direct sum over an empty type is trivial.

Equations

DirectSum.uniqueOfIsEmpty = DFinsupp.uniqueOfIsEmpty

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def DirectSum.id (M : Type v) (ι : optParam (Type u_1) PUnit.{u_1 + 1} ) [AddCommMonoid M] [Unique ι] :

(DirectSum ι fun (x : ι) => M) ≃+ M

The natural equivalence between ⨁ _ : ι, M and M when Unique ι.

Equations

One or more equations did not get rendered due to their size.

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def DirectSum.equivCongrLeft {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {κ : Type u_1} (h : ι ≃ κ) :

(DirectSum ι fun (i : ι) => β i) ≃+ DirectSum κ fun (k : κ) => β (h.symm k)

Reindexing terms of a direct sum.

Equations

DirectSum.equivCongrLeft h = let __src := DFinsupp.equivCongrLeft h; { toEquiv := __src, map_add' := ⋯ }

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@[simp]

theorem DirectSum.equivCongrLeft_apply {ι : Type v} {β : ι → Type w} [(i : ι) → AddCommMonoid (β i)] {κ : Type u_1} (h : ι ≃ κ) (f : DirectSum ι fun (i : ι) => β i) (k : κ) :

((DirectSum.equivCongrLeft h) f) k = f (h.symm k)

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@[simp]

theorem DirectSum.addEquivProdDirectSum_apply {ι : Type v} [dec_ι : DecidableEq ι] {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] (f : Π₀ (i : Option ι), (fun (i : Option ι) => (fun (i : Option ι) => α i) i) i) :

DirectSum.addEquivProdDirectSum f = (f none, DFinsupp.comapDomain some ⋯ f)

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@[simp]

theorem DirectSum.addEquivProdDirectSum_symm_apply_toFun {ι : Type v} [dec_ι : DecidableEq ι] {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] (f : (fun (i : Option ι) => (fun (i : Option ι) => α i) i) none × Π₀ (i : ι), (fun (i : Option ι) => (fun (i : Option ι) => α i) i) (some i)) (i : Option ι) :

((AddEquiv.symm DirectSum.addEquivProdDirectSum) f) i = match i with | none => f.1 | some val => f.2 val

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@[simp]

theorem DirectSum.addEquivProdDirectSum_symm_apply_support' {ι : Type v} [dec_ι : DecidableEq ι] {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] (f : (fun (i : Option ι) => (fun (i : Option ι) => α i) i) none × Π₀ (i : ι), (fun (i : Option ι) => (fun (i : Option ι) => α i) i) (some i)) :

((AddEquiv.symm DirectSum.addEquivProdDirectSum) f).support' = Trunc.map (fun (s : { s : Multiset ι // ∀ (i : ι), i ∈ s ∨ f.2 i = 0 }) => { val := none ::ₘ Multiset.map some ↑s, property := ⋯ }) f.2.support'

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noncomputable def DirectSum.addEquivProdDirectSum {ι : Type v} [dec_ι : DecidableEq ι] {α : Option ι → Type w} [(i : Option ι) → AddCommMonoid (α i)] :

(DirectSum (Option ι) fun (i : Option ι) => α i) ≃+ α none × DirectSum ι fun (i : ι) => α (some i)

Isomorphism obtained by separating the term of index none of a direct sum over Option ι.

Equations

DirectSum.addEquivProdDirectSum = let __src := DFinsupp.equivProdDFinsupp; { toEquiv := __src, map_add' := ⋯ }

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def DirectSum.sigmaCurry {ι : Type v} [dec_ι : DecidableEq ι] {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] :

(DirectSum ((_i : ι) × α _i) fun (i : (_i : ι) × α _i) => δ i.fst i.snd) →+ DirectSum ι fun (i : ι) => DirectSum (α i) fun (j : α i) => δ i j

The natural map between ⨁ (i : Σ i, α i), δ i.1 i.2 and ⨁ i (j : α i), δ i j.

Equations

DirectSum.sigmaCurry = { toZeroHom := { toFun := DFinsupp.sigmaCurry, map_zero' := ⋯ }, map_add' := ⋯ }

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@[simp]

theorem DirectSum.sigmaCurry_apply {ι : Type v} [dec_ι : DecidableEq ι] {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] (f : DirectSum ((_i : ι) × α _i) fun (i : (_i : ι) × α _i) => δ i.fst i.snd) (i : ι) (j : α i) :

((DirectSum.sigmaCurry f) i) j = f { fst := i, snd := j }

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def DirectSum.sigmaUncurry {ι : Type v} {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] [(i : ι) → DecidableEq (α i)] [(i : ι) → (j : α i) → DecidableEq (δ i j)] :

(DirectSum ι fun (i : ι) => DirectSum (α i) fun (j : α i) => δ i j) →+ DirectSum ((_i : ι) × α _i) fun (i : (_i : ι) × α _i) => δ i.fst i.snd

The natural map between ⨁ i (j : α i), δ i j and Π₀ (i : Σ i, α i), δ i.1 i.2, inverse of curry.

Equations

DirectSum.sigmaUncurry = { toZeroHom := { toFun := DFinsupp.sigmaUncurry, map_zero' := ⋯ }, map_add' := ⋯ }

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@[simp]

theorem DirectSum.sigmaUncurry_apply {ι : Type v} {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] [(i : ι) → DecidableEq (α i)] [(i : ι) → (j : α i) → DecidableEq (δ i j)] (f : DirectSum ι fun (i : ι) => DirectSum (α i) fun (j : α i) => δ i j) (i : ι) (j : α i) :

(DirectSum.sigmaUncurry f) { fst := i, snd := j } = (f i) j

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def DirectSum.sigmaCurryEquiv {ι : Type v} [dec_ι : DecidableEq ι] {α : ι → Type u} {δ : (i : ι) → α i → Type w} [(i : ι) → (j : α i) → AddCommMonoid (δ i j)] [(i : ι) → DecidableEq (α i)] [(i : ι) → (j : α i) → DecidableEq (δ i j)] :

(DirectSum ((_i : ι) × α _i) fun (i : (_i : ι) × α _i) => δ i.fst i.snd) ≃+ DirectSum ι fun (i : ι) => DirectSum (α i) fun (j : α i) => δ i j

The natural map between ⨁ (i : Σ i, α i), δ i.1 i.2 and ⨁ i (j : α i), δ i j.

Equations

One or more equations did not get rendered due to their size.

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def DirectSum.coeAddMonoidHom {ι : Type v} {M : Type u_1} {S : Type u_2} [DecidableEq ι] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] (A : ι → S) :

(DirectSum ι fun (i : ι) => ↥(A i)) →+ M

The canonical embedding from ⨁ i, A i to M where A is a collection of AddSubmonoid M indexed by ι.

When S = Submodule _ M, this is available as a LinearMap, DirectSum.coe_linearMap.

Equations

DirectSum.coeAddMonoidHom A = DirectSum.toAddMonoid fun (i : ι) => AddSubmonoidClass.subtype (A i)

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@[simp]

theorem DirectSum.coeAddMonoidHom_of {ι : Type v} {M : Type u_1} {S : Type u_2} [DecidableEq ι] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] (A : ι → S) (i : ι) (x : ↥(A i)) :

(DirectSum.coeAddMonoidHom A) ((DirectSum.of (fun (i : ι) => ↥(A i)) i) x) = ↑x

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theorem DirectSum.coe_of_apply {ι : Type v} {M : Type u_1} {S : Type u_2} [DecidableEq ι] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] {A : ι → S} (i : ι) (j : ι) (x : ↥(A i)) :

↑(((DirectSum.of (fun (i : ι) => ↥(A i)) i) x) j) = ↑(if i = j then x else 0)

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def DirectSum.IsInternal {ι : Type v} {M : Type u_1} {S : Type u_2} [DecidableEq ι] [AddCommMonoid M] [SetLike S M] [AddSubmonoidClass S M] (A : ι → S) :

Prop

The DirectSum formed by a collection of additive submonoids (or subgroups, or submodules) of M is said to be internal if the canonical map (⨁ i, A i) →+ M is bijective.

For the alternate statement in terms of independence and spanning, see DirectSum.subgroup_isInternal_iff_independent_and_supr_eq_top and DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top.

Equations

DirectSum.IsInternal A = Function.Bijective ⇑(DirectSum.coeAddMonoidHom A)

source

theorem DirectSum.IsInternal.addSubmonoid_iSup_eq_top {ι : Type v} {M : Type u_1} [DecidableEq ι] [AddCommMonoid M] (A : ι → AddSubmonoid M) (h : DirectSum.IsInternal A) :

iSup A = ⊤

Documentation

Mathlib.Algebra.DirectSum.Basic

Direct sum #

Notation #

References #