Direct limit of modules, abelian groups, rings, and fields.

See Atiyah-Macdonald PP.32-33, Matsumura PP.269-270

Generalizes the notion of "union", or "gluing", of incomparable modules over the same ring, or incomparable abelian groups, or rings, or fields.

It is constructed as a quotient of the free module (for the module case) or quotient of the free commutative ring (for the ring case) instead of a quotient of the disjoint union so as to make the operations (addition etc.) "computable".

Main definitions

• DirectedSystem f
• Module.DirectLimit G f
• AddCommGroup.DirectLimit G f
• Ring.DirectLimit G f

class DirectedSystem {ι : Type v} [Preorder ι] (G : ι → Type w) (f : (i j : ι) → i ≤ j → G i → G j) : Prop

A directed system is a functor from a category (directed poset) to another category.

• map_self' : ∀ (i : ι) (x : G i) (h : i ≤ i), f i i h x = x
• map_map' : ∀ {i j k : ι} (hij : i ≤ j) (hjk : j ≤ k) (x : G i), f j k hjk (f i j hij x) = f i k ⋯ x

theorem DirectedSystem.map_self {ι : Type v} [Preorder ι] {G : ι → Type w} (f : (i j : ι) → i ≤ j → G i → G j) [DirectedSystem G fun (i j : ι) (h : i ≤ j) => f i j h] (i : ι) (x : G i) (h : i ≤ i) : f i i h x = x

theorem DirectedSystem.map_map {ι : Type v} [Preorder ι] {G : ι → Type w} (f : (i j : ι) → i ≤ j → G i → G j) [DirectedSystem G fun (i j : ι) (h : i ≤ j) => f i j h] {i : ι} {j : ι} {k : ι} (hij : i ≤ j) (hjk : j ≤ k) (x : G i) : f j k hjk (f i j hij x) = f i k ⋯ x

theorem Module.DirectedSystem.map_self {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f i j h)] (i : ι) (x : G i) (h : i ≤ i) : (f i i h) x = x

A copy of DirectedSystem.map_self specialized to linear maps, as otherwise the fun i j h ↦ f i j h can confuse the simplifier.

theorem Module.DirectedSystem.map_map {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f i j h)] {i : ι} {j : ι} {k : ι} (hij : i ≤ j) (hjk : j ≤ k) (x : G i) : (f j k hjk) ((f i j hij) x) = (f i k ⋯) x

A copy of DirectedSystem.map_map specialized to linear maps, as otherwise the fun i j h ↦ f i j h can confuse the simplifier.

noncomputable def Module.DirectLimit {R : Type u} [Ring R] {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) [DecidableEq ι] : Type (max v w)

The direct limit of a directed system is the modules glued together along the maps.

Equations

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

noncomputable instance Module.DirectLimit.addCommGroup {R : Type u} [Ring R] {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) [DecidableEq ι] : AddCommGroup (Module.DirectLimit G f)

Equations

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

noncomputable instance Module.DirectLimit.module {R : Type u} [Ring R] {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) [DecidableEq ι] : Module R (Module.DirectLimit G f)

Equations

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

noncomputable instance Module.DirectLimit.inhabited {R : Type u} [Ring R] {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) [DecidableEq ι] : Inhabited (Module.DirectLimit G f)

Equations

• Module.DirectLimit.inhabited G f = { default := 0 }

noncomputable instance Module.DirectLimit.unique {R : Type u} [Ring R] {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) [DecidableEq ι] [IsEmpty ι] : Unique (Module.DirectLimit G f)

Equations

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

noncomputable def Module.DirectLimit.of (R : Type u) [Ring R] (ι : Type v) [Preorder ι] (G : ι → Type w) [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) [DecidableEq ι] (i : ι) : G i →ₗ[R] Module.DirectLimit G f

The canonical map from a component to the direct limit.

Equations

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

@[simp]

theorem Module.DirectLimit.of_f {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {i : ι} {j : ι} {hij : i ≤ j} {x : G i} : (Module.DirectLimit.of R ι G f j) ((f i j hij) x) = (Module.DirectLimit.of R ι G f i) x

theorem Module.DirectLimit.exists_of {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (z : Module.DirectLimit G f) : ∃ (i : ι) (x : G i), (Module.DirectLimit.of R ι G f i) x = z

Every element of the direct limit corresponds to some element in some component of the directed system.

theorem Module.DirectLimit.induction_on {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] {C : Module.DirectLimit G f → Prop} (z : Module.DirectLimit G f) (ih : ∀ (i : ι) (x : G i), C ((Module.DirectLimit.of R ι G f i) x)) : C z

noncomputable def Module.DirectLimit.lift (R : Type u) [Ring R] (ι : Type v) [Preorder ι] (G : ι → Type w) [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) [DecidableEq ι] {P : Type u₁} [AddCommGroup P] [Module R P] (g : (i : ι) → G i →ₗ[R] P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) : Module.DirectLimit G f →ₗ[R] P

The universal property of the direct limit: maps from the components to another module that respect the directed system structure (i.e. make some diagram commute) give rise to a unique map out of the direct limit.

Equations

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

theorem Module.DirectLimit.lift_of {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {P : Type u₁} [AddCommGroup P] [Module R P] (g : (i : ι) → G i →ₗ[R] P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) {i : ι} (x : G i) : (Module.DirectLimit.lift R ι G f g Hg) ((Module.DirectLimit.of R ι G f i) x) = (g i) x

theorem Module.DirectLimit.lift_unique {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {P : Type u₁} [AddCommGroup P] [Module R P] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (F : Module.DirectLimit G f →ₗ[R] P) (x : Module.DirectLimit G f) : F x = (Module.DirectLimit.lift R ι G f (fun (i : ι) => F ∘ₗ Module.DirectLimit.of R ι G f i) ⋯) x

theorem Module.DirectLimit.lift_injective {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {P : Type u₁} [AddCommGroup P] [Module R P] (g : (i : ι) → G i →ₗ[R] P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (injective : ∀ (i : ι), Function.Injective ⇑(g i)) : Function.Injective ⇑(Module.DirectLimit.lift R ι G f g Hg)

noncomputable def Module.DirectLimit.map {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] [(i : ι) → Module R (G' i)] {f' : (i j : ι) → i ≤ j → G' i →ₗ[R] G' j} (g : (i : ι) → G i →ₗ[R] G' i) (hg : ∀ (i j : ι) (h : i ≤ j), g j ∘ₗ f i j h = f' i j h ∘ₗ g i) : Module.DirectLimit G f →ₗ[R] Module.DirectLimit G' f'

Consider direct limits lim G and lim G' with direct system f and f' respectively, any family of linear maps gᵢ : Gᵢ ⟶ G'ᵢ such that g ∘ f = f' ∘ g induces a linear map lim G ⟶ lim G'.

Equations

• Module.DirectLimit.map g hg = Module.DirectLimit.lift R ι G f (fun (i : ι) => LinearMap.comp (Module.DirectLimit.of R ι G' f' i) (g i)) ⋯

@[simp]

theorem Module.DirectLimit.map_apply_of {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] [(i : ι) → Module R (G' i)] {f' : (i j : ι) → i ≤ j → G' i →ₗ[R] G' j} (g : (i : ι) → G i →ₗ[R] G' i) (hg : ∀ (i j : ι) (h : i ≤ j), g j ∘ₗ f i j h = f' i j h ∘ₗ g i) {i : ι} (x : G i) : (Module.DirectLimit.map g hg) ((Module.DirectLimit.of R ι G f i) x) = (Module.DirectLimit.of R ι G' f' i) ((g i) x)

@[simp]

theorem Module.DirectLimit.map_id {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] : Module.DirectLimit.map (fun (i : ι) => LinearMap.id) ⋯ = LinearMap.id

theorem Module.DirectLimit.map_comp {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] [(i : ι) → Module R (G' i)] {f' : (i j : ι) → i ≤ j → G' i →ₗ[R] G' j} {G'' : ι → Type v''} [(i : ι) → AddCommGroup (G'' i)] [(i : ι) → Module R (G'' i)] {f'' : (i j : ι) → i ≤ j → G'' i →ₗ[R] G'' j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (g₁ : (i : ι) → G i →ₗ[R] G' i) (g₂ : (i : ι) → G' i →ₗ[R] G'' i) (hg₁ : ∀ (i j : ι) (h : i ≤ j), g₁ j ∘ₗ f i j h = f' i j h ∘ₗ g₁ i) (hg₂ : ∀ (i j : ι) (h : i ≤ j), g₂ j ∘ₗ f' i j h = f'' i j h ∘ₗ g₂ i) : Module.DirectLimit.map g₂ hg₂ ∘ₗ Module.DirectLimit.map g₁ hg₁ = Module.DirectLimit.map (fun (i : ι) => g₂ i ∘ₗ g₁ i) ⋯

noncomputable def Module.DirectLimit.congr {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] [(i : ι) → Module R (G' i)] {f' : (i j : ι) → i ≤ j → G' i →ₗ[R] G' j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (e : (i : ι) → G i ≃ₗ[R] G' i) (he : ∀ (i j : ι) (h : i ≤ j), ↑(e j) ∘ₗ f i j h = f' i j h ∘ₗ ↑(e i)) : Module.DirectLimit G f ≃ₗ[R] Module.DirectLimit G' f'

Consider direct limits lim G and lim G' with direct system f and f' respectively, any family of equivalences eᵢ : Gᵢ ≅ G'ᵢ such that e ∘ f = f' ∘ e induces an equivalence lim G ≅ lim G'.

Equations

• Module.DirectLimit.congr e he = LinearEquiv.ofLinear (Module.DirectLimit.map (fun (x : ι) => ↑(e x)) he) (Module.DirectLimit.map (fun (i : ι) => ↑(LinearEquiv.symm (e i))) ⋯) ⋯ ⋯

theorem Module.DirectLimit.congr_apply_of {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] [(i : ι) → Module R (G' i)] {f' : (i j : ι) → i ≤ j → G' i →ₗ[R] G' j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (e : (i : ι) → G i ≃ₗ[R] G' i) (he : ∀ (i j : ι) (h : i ≤ j), ↑(e j) ∘ₗ f i j h = f' i j h ∘ₗ ↑(e i)) {i : ι} (g : G i) : (Module.DirectLimit.congr e he) ((Module.DirectLimit.of R ι G f i) g) = (Module.DirectLimit.of R ι G' f' i) ((e i) g)

theorem Module.DirectLimit.congr_symm_apply_of {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] [(i : ι) → Module R (G' i)] {f' : (i j : ι) → i ≤ j → G' i →ₗ[R] G' j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (e : (i : ι) → G i ≃ₗ[R] G' i) (he : ∀ (i j : ι) (h : i ≤ j), ↑(e j) ∘ₗ f i j h = f' i j h ∘ₗ ↑(e i)) {i : ι} (g : G' i) : (LinearEquiv.symm (Module.DirectLimit.congr e he)) ((Module.DirectLimit.of R ι G' f' i) g) = (Module.DirectLimit.of R ι G f i) ((LinearEquiv.symm (e i)) g)

noncomputable def Module.DirectLimit.totalize {R : Type u} [Ring R] {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) (i : ι) (j : ι) : G i →ₗ[R] G j

totalize G f i j is a linear map from G i to G j, for every i and j. If i ≤ j, then it is the map f i j that comes with the directed system G, and otherwise it is the zero map.

Equations

• Module.DirectLimit.totalize G f i j = if h : i ≤ j then f i j h else 0

theorem Module.DirectLimit.totalize_of_le {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} {i : ι} {j : ι} (h : i ≤ j) : Module.DirectLimit.totalize G f i j = f i j h

theorem Module.DirectLimit.totalize_of_not_le {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} {i : ι} {j : ι} (h : ¬i ≤ j) : Module.DirectLimit.totalize G f i j = 0

theorem Module.DirectLimit.toModule_totalize_of_le {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f i j h)] [(i : ι) → (k : G i) → Decidable (k ≠ 0)] {x : DirectSum ι G} {i : ι} {j : ι} (hij : i ≤ j) (hx : ∀ k ∈ DFinsupp.support x, k ≤ i) : (DirectSum.toModule R ι (G j) fun (k : ι) => Module.DirectLimit.totalize G f k j) x = (f i j hij) ((DirectSum.toModule R ι (G i) fun (k : ι) => Module.DirectLimit.totalize G f k i) x)

theorem Module.DirectLimit.of.zero_exact_aux {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f i j h)] [(i : ι) → (k : G i) → Decidable (k ≠ 0)] [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] {x : DirectSum ι G} (H : Submodule.Quotient.mk x = 0) : ∃ (j : ι), (∀ k ∈ DFinsupp.support x, k ≤ j) ∧ (DirectSum.toModule R ι (G j) fun (i : ι) => Module.DirectLimit.totalize G f i j) x = 0

theorem Module.DirectLimit.of.zero_exact {R : Type u} [Ring R] {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] [(i : ι) → Module R (G i)] {f : (i j : ι) → i ≤ j → G i →ₗ[R] G j} [DecidableEq ι] [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f i j h)] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] {i : ι} {x : G i} (H : (Module.DirectLimit.of R ι G f i) x = 0) : ∃ (j : ι) (hij : i ≤ j), (f i j hij) x = 0

A component that corresponds to zero in the direct limit is already zero in some bigger module in the directed system.

noncomputable def AddCommGroup.DirectLimit {ι : Type v} [Preorder ι] (G : ι → Type w) [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] (f : (i j : ι) → i ≤ j → G i →+ G j) : Type (max w v)

The direct limit of a directed system is the abelian groups glued together along the maps.

Equations

• AddCommGroup.DirectLimit G f = Module.DirectLimit G fun (i j : ι) (hij : i ≤ j) => AddMonoidHom.toIntLinearMap (f i j hij)

theorem AddCommGroup.DirectLimit.directedSystem {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → AddCommGroup (G i)] (f : (i j : ι) → i ≤ j → G i →+ G j) [h : DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f i j h)] : DirectedSystem G fun (i j : ι) (hij : i ≤ j) => ⇑(AddMonoidHom.toIntLinearMap (f i j hij))

noncomputable instance AddCommGroup.DirectLimit.instAddCommGroupDirectLimit {ι : Type v} [Preorder ι] (G : ι → Type w) [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] (f : (i j : ι) → i ≤ j → G i →+ G j) : AddCommGroup (AddCommGroup.DirectLimit G f)

Equations

• AddCommGroup.DirectLimit.instAddCommGroupDirectLimit G f = Module.DirectLimit.addCommGroup G fun (i j : ι) (hij : i ≤ j) => AddMonoidHom.toIntLinearMap (f i j hij)

noncomputable instance AddCommGroup.DirectLimit.instInhabitedDirectLimit {ι : Type v} [Preorder ι] (G : ι → Type w) [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] (f : (i j : ι) → i ≤ j → G i →+ G j) : Inhabited (AddCommGroup.DirectLimit G f)

Equations

• AddCommGroup.DirectLimit.instInhabitedDirectLimit G f = { default := 0 }

noncomputable instance AddCommGroup.DirectLimit.instUniqueDirectLimit {ι : Type v} [Preorder ι] (G : ι → Type w) [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] (f : (i j : ι) → i ≤ j → G i →+ G j) [IsEmpty ι] : Unique (AddCommGroup.DirectLimit G f)

Equations

• AddCommGroup.DirectLimit.instUniqueDirectLimit G f = Module.DirectLimit.unique G fun (i j : ι) (hij : i ≤ j) => AddMonoidHom.toIntLinearMap (f i j hij)

noncomputable def AddCommGroup.DirectLimit.of {ι : Type v} [Preorder ι] (G : ι → Type w) [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] (f : (i j : ι) → i ≤ j → G i →+ G j) (i : ι) : G i →+ AddCommGroup.DirectLimit G f

The canonical map from a component to the direct limit.

Equations

• AddCommGroup.DirectLimit.of G f i = LinearMap.toAddMonoidHom (Module.DirectLimit.of ℤ ι G (fun (i j : ι) (hij : i ≤ j) => AddMonoidHom.toIntLinearMap (f i j hij)) i)

@[simp]

theorem AddCommGroup.DirectLimit.of_f {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} {i : ι} {j : ι} (hij : i ≤ j) (x : G i) : (AddCommGroup.DirectLimit.of G f j) ((f i j hij) x) = (AddCommGroup.DirectLimit.of G f i) x

theorem AddCommGroup.DirectLimit.induction_on {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] {C : AddCommGroup.DirectLimit G f → Prop} (z : AddCommGroup.DirectLimit G f) (ih : ∀ (i : ι) (x : G i), C ((AddCommGroup.DirectLimit.of G f i) x)) : C z

theorem AddCommGroup.DirectLimit.of.zero_exact {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f i j h)] (i : ι) (x : G i) (h : (AddCommGroup.DirectLimit.of G f i) x = 0) : ∃ (j : ι) (hij : i ≤ j), (f i j hij) x = 0

A component that corresponds to zero in the direct limit is already zero in some bigger module in the directed system.

noncomputable def AddCommGroup.DirectLimit.lift {ι : Type v} [Preorder ι] (G : ι → Type w) [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] (f : (i j : ι) → i ≤ j → G i →+ G j) (P : Type u₁) [AddCommGroup P] (g : (i : ι) → G i →+ P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) : AddCommGroup.DirectLimit G f →+ P

The universal property of the direct limit: maps from the components to another abelian group that respect the directed system structure (i.e. make some diagram commute) give rise to a unique map out of the direct limit.

Equations

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

@[simp]

theorem AddCommGroup.DirectLimit.lift_of {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} (P : Type u₁) [AddCommGroup P] (g : (i : ι) → G i →+ P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) (i : ι) (x : G i) : (AddCommGroup.DirectLimit.lift G f P g Hg) ((AddCommGroup.DirectLimit.of G f i) x) = (g i) x

theorem AddCommGroup.DirectLimit.lift_unique {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} (P : Type u₁) [AddCommGroup P] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (F : AddCommGroup.DirectLimit G f →+ P) (x : AddCommGroup.DirectLimit G f) : F x = (AddCommGroup.DirectLimit.lift G f P (fun (i : ι) => AddMonoidHom.comp F (AddCommGroup.DirectLimit.of G f i)) ⋯) x

theorem AddCommGroup.DirectLimit.lift_injective {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} (P : Type u₁) [AddCommGroup P] (g : (i : ι) → G i →+ P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (injective : ∀ (i : ι), Function.Injective ⇑(g i)) : Function.Injective ⇑(AddCommGroup.DirectLimit.lift G f P g Hg)

noncomputable def AddCommGroup.DirectLimit.map {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} (g : (i : ι) → G i →+ G' i) (hg : ∀ (i j : ι) (h : i ≤ j), AddMonoidHom.comp (g j) (f i j h) = AddMonoidHom.comp (f' i j h) (g i)) : AddCommGroup.DirectLimit G f →+ AddCommGroup.DirectLimit G' f'

Consider direct limits lim G and lim G' with direct system f and f' respectively, any family of group homomorphisms gᵢ : Gᵢ ⟶ G'ᵢ such that g ∘ f = f' ∘ g induces a group homomorphism lim G ⟶ lim G'.

Equations

• AddCommGroup.DirectLimit.map g hg = AddCommGroup.DirectLimit.lift G f (AddCommGroup.DirectLimit G' f') (fun (i : ι) => AddMonoidHom.comp (AddCommGroup.DirectLimit.of G' f' i) (g i)) ⋯

@[simp]

theorem AddCommGroup.DirectLimit.map_apply_of {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} (g : (i : ι) → G i →+ G' i) (hg : ∀ (i j : ι) (h : i ≤ j), AddMonoidHom.comp (g j) (f i j h) = AddMonoidHom.comp (f' i j h) (g i)) {i : ι} (x : G i) : (AddCommGroup.DirectLimit.map g hg) ((AddCommGroup.DirectLimit.of G f i) x) = (AddCommGroup.DirectLimit.of G' f' i) ((g i) x)

@[simp]

theorem AddCommGroup.DirectLimit.map_id {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] : AddCommGroup.DirectLimit.map (fun (i : ι) => AddMonoidHom.id (G i)) ⋯ = AddMonoidHom.id (AddCommGroup.DirectLimit G f)

theorem AddCommGroup.DirectLimit.map_comp {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} {G'' : ι → Type v''} [(i : ι) → AddCommGroup (G'' i)] {f'' : (i j : ι) → i ≤ j → G'' i →+ G'' j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (g₁ : (i : ι) → G i →+ G' i) (g₂ : (i : ι) → G' i →+ G'' i) (hg₁ : ∀ (i j : ι) (h : i ≤ j), AddMonoidHom.comp (g₁ j) (f i j h) = AddMonoidHom.comp (f' i j h) (g₁ i)) (hg₂ : ∀ (i j : ι) (h : i ≤ j), AddMonoidHom.comp (g₂ j) (f' i j h) = AddMonoidHom.comp (f'' i j h) (g₂ i)) : AddMonoidHom.comp (AddCommGroup.DirectLimit.map g₂ hg₂) (AddCommGroup.DirectLimit.map g₁ hg₁) = AddCommGroup.DirectLimit.map (fun (i : ι) => AddMonoidHom.comp (g₂ i) (g₁ i)) ⋯

noncomputable def AddCommGroup.DirectLimit.congr {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (e : (i : ι) → G i ≃+ G' i) (he : ∀ (i j : ι) (h : i ≤ j), AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)) : AddCommGroup.DirectLimit G f ≃+ AddCommGroup.DirectLimit G' f'

Consider direct limits lim G and lim G' with direct system f and f' respectively, any family of equivalences eᵢ : Gᵢ ≅ G'ᵢ such that e ∘ f = f' ∘ e induces an equivalence lim G ⟶ lim G'.

Equations

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

theorem AddCommGroup.DirectLimit.congr_apply_of {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (e : (i : ι) → G i ≃+ G' i) (he : ∀ (i j : ι) (h : i ≤ j), AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)) {i : ι} (g : G i) : (AddCommGroup.DirectLimit.congr e he) ((AddCommGroup.DirectLimit.of G f i) g) = (AddCommGroup.DirectLimit.of G' f' i) ((e i) g)

theorem AddCommGroup.DirectLimit.congr_symm_apply_of {ι : Type v} [Preorder ι] {G : ι → Type w} [DecidableEq ι] [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} {G' : ι → Type v'} [(i : ι) → AddCommGroup (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+ G' j} [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (e : (i : ι) → G i ≃+ G' i) (he : ∀ (i j : ι) (h : i ≤ j), AddMonoidHom.comp (AddEquiv.toAddMonoidHom (e j)) (f i j h) = AddMonoidHom.comp (f' i j h) ↑(e i)) {i : ι} (g : G' i) : (AddEquiv.symm (AddCommGroup.DirectLimit.congr e he)) ((AddCommGroup.DirectLimit.of G' f' i) g) = (AddCommGroup.DirectLimit.of G f i) ((AddEquiv.symm (e i)) g)

noncomputable def Ring.DirectLimit {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → CommRing (G i)] (f : (i j : ι) → i ≤ j → G i → G j) : Type (max v w)

The direct limit of a directed system is the rings glued together along the maps.

Equations

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

noncomputable instance Ring.DirectLimit.commRing {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → CommRing (G i)] (f : (i j : ι) → i ≤ j → G i → G j) : CommRing (Ring.DirectLimit G f)

Equations

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

noncomputable instance Ring.DirectLimit.ring {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → CommRing (G i)] (f : (i j : ι) → i ≤ j → G i → G j) : Ring (Ring.DirectLimit G f)

Equations

• Ring.DirectLimit.ring G f = CommRing.toRing

noncomputable instance Ring.DirectLimit.zero {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → CommRing (G i)] (f : (i j : ι) → i ≤ j → G i → G j) : Zero (Ring.DirectLimit G f)

Equations

• Ring.DirectLimit.zero G f = id { zero := 0 }

noncomputable instance Ring.DirectLimit.instInhabitedDirectLimit {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → CommRing (G i)] (f : (i j : ι) → i ≤ j → G i → G j) : Inhabited (Ring.DirectLimit G f)

Equations

• Ring.DirectLimit.instInhabitedDirectLimit G f = { default := 0 }

noncomputable def Ring.DirectLimit.of {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → CommRing (G i)] (f : (i j : ι) → i ≤ j → G i → G j) (i : ι) : G i →+* Ring.DirectLimit G f

The canonical map from a component to the direct limit.

Equations

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

theorem Ring.DirectLimit.of_f {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i → G j} {i : ι} {j : ι} (hij : i ≤ j) (x : G i) : (Ring.DirectLimit.of G f j) (f i j hij x) = (Ring.DirectLimit.of G f i) x

theorem Ring.DirectLimit.exists_of {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i → G j} [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (z : Ring.DirectLimit G f) : ∃ (i : ι) (x : G i), (Ring.DirectLimit.of G f i) x = z

Every element of the direct limit corresponds to some element in some component of the directed system.

theorem Ring.DirectLimit.Polynomial.exists_of {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f' : (i j : ι) → i ≤ j → G i →+* G j} [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (q : Polynomial (Ring.DirectLimit G fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h))) : ∃ (i : ι) (p : Polynomial (G i)), Polynomial.map (Ring.DirectLimit.of G (fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)) i) p = q

theorem Ring.DirectLimit.induction_on {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i → G j} [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] {C : Ring.DirectLimit G f → Prop} (z : Ring.DirectLimit G f) (ih : ∀ (i : ι) (x : G i), C ((Ring.DirectLimit.of G f i) x)) : C z

theorem Ring.DirectLimit.of.zero_exact_aux2 {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → CommRing (G i)] (f' : (i j : ι) → i ≤ j → G i →+* G j) [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)] {x : FreeCommRing ((i : ι) × G i)} {s : Set ((i : ι) × G i)} {t : Set ((i : ι) × G i)} [DecidablePred fun (x : (i : ι) × G i) => x ∈ s] [DecidablePred fun (x : (i : ι) × G i) => x ∈ t] (hxs : FreeCommRing.IsSupported x s) {j : ι} {k : ι} (hj : ∀ z ∈ s, z.fst ≤ j) (hk : ∀ z ∈ t, z.fst ≤ k) (hjk : j ≤ k) (hst : s ⊆ t) : (f' j k hjk) ((FreeCommRing.lift fun (ix : ↑s) => (f' (↑ix).fst j ⋯) (↑ix).snd) ((FreeCommRing.restriction s) x)) = (FreeCommRing.lift fun (ix : ↑t) => (f' (↑ix).fst k ⋯) (↑ix).snd) ((FreeCommRing.restriction t) x)

theorem Ring.DirectLimit.of.zero_exact_aux {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f' : (i j : ι) → i ≤ j → G i →+* G j} [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)] [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] {x : FreeCommRing ((i : ι) × G i)} (H : (Ideal.Quotient.mk (Ideal.span {a : FreeCommRing ((i : ι) × G i) | (∃ (i : ι) (j : ι) (H : i ≤ j) (x : G i), FreeCommRing.of { fst := j, snd := (fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)) i j H x } - FreeCommRing.of { fst := i, snd := x } = a) ∨ (∃ (i : ι), FreeCommRing.of { fst := i, snd := 1 } - 1 = a) ∨ (∃ (i : ι) (x : G i) (y : G i), FreeCommRing.of { fst := i, snd := x + y } - (FreeCommRing.of { fst := i, snd := x } + FreeCommRing.of { fst := i, snd := y }) = a) ∨ ∃ (i : ι) (x : G i) (y : G i), FreeCommRing.of { fst := i, snd := x * y } - FreeCommRing.of { fst := i, snd := x } * FreeCommRing.of { fst := i, snd := y } = a})) x = 0) : ∃ (j : ι) (s : Set ((i : ι) × G i)) (H : ∀ k ∈ s, k.fst ≤ j), FreeCommRing.IsSupported x s ∧ ∀ [inst : DecidablePred fun (x : (i : ι) × G i) => x ∈ s], (FreeCommRing.lift fun (ix : ↑s) => (f' (↑ix).fst j ⋯) (↑ix).snd) ((FreeCommRing.restriction s) x) = 0

theorem Ring.DirectLimit.of.zero_exact {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f' : (i j : ι) → i ≤ j → G i →+* G j} [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] {i : ι} {x : G i} (hix : (Ring.DirectLimit.of G (fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)) i) x = 0) : ∃ (j : ι) (hij : i ≤ j), (f' i j hij) x = 0

A component that corresponds to zero in the direct limit is already zero in some bigger module in the directed system.

theorem Ring.DirectLimit.of_injective {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] (f' : (i j : ι) → i ≤ j → G i →+* G j) [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)] (hf : ∀ (i j : ι) (hij : i ≤ j), Function.Injective ⇑(f' i j hij)) (i : ι) : Function.Injective ⇑(Ring.DirectLimit.of G (fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)) i)

If the maps in the directed system are injective, then the canonical maps from the components to the direct limits are injective.

noncomputable def Ring.DirectLimit.lift {ι : Type v} [Preorder ι] (G : ι → Type w) [(i : ι) → CommRing (G i)] (f : (i j : ι) → i ≤ j → G i → G j) (P : Type u₁) [CommRing P] (g : (i : ι) → G i →+* P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f i j hij x) = (g i) x) : Ring.DirectLimit G f →+* P

The universal property of the direct limit: maps from the components to another ring that respect the directed system structure (i.e. make some diagram commute) give rise to a unique map out of the direct limit.

Equations

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

theorem Ring.DirectLimit.lift_of {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i → G j} (P : Type u₁) [CommRing P] (g : (i : ι) → G i →+* P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f i j hij x) = (g i) x) (i : ι) (x : G i) : (Ring.DirectLimit.lift G f P g Hg) ((Ring.DirectLimit.of G f i) x) = (g i) x

theorem Ring.DirectLimit.lift_unique {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i → G j} (P : Type u₁) [CommRing P] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (F : Ring.DirectLimit G f →+* P) (x : Ring.DirectLimit G f) : F x = (Ring.DirectLimit.lift G f P (fun (i : ι) => RingHom.comp F (Ring.DirectLimit.of G f i)) ⋯) x

theorem Ring.DirectLimit.lift_injective {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i → G j} (P : Type u₁) [CommRing P] (g : (i : ι) → G i →+* P) (Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) (f i j hij x) = (g i) x) [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (injective : ∀ (i : ι), Function.Injective ⇑(g i)) : Function.Injective ⇑(Ring.DirectLimit.lift G f P g Hg)

noncomputable def Ring.DirectLimit.map {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type v'} [(i : ι) → CommRing (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+* G' j} (g : (i : ι) → G i →+* G' i) (hg : ∀ (i j : ι) (h : i ≤ j), RingHom.comp (g j) (f i j h) = RingHom.comp (f' i j h) (g i)) : (Ring.DirectLimit G fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f x x_1 h)) →+* Ring.DirectLimit G' fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f' x x_1 h)

Consider direct limits lim G and lim G' with direct system f and f' respectively, any family of ring homomorphisms gᵢ : Gᵢ ⟶ G'ᵢ such that g ∘ f = f' ∘ g induces a ring homomorphism lim G ⟶ lim G'.

Equations

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

@[simp]

theorem Ring.DirectLimit.map_apply_of {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type v'} [(i : ι) → CommRing (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+* G' j} (g : (i : ι) → G i →+* G' i) (hg : ∀ (i j : ι) (h : i ≤ j), RingHom.comp (g j) (f i j h) = RingHom.comp (f' i j h) (g i)) {i : ι} (x : G i) : (Ring.DirectLimit.map g hg) ((Ring.DirectLimit.of G (fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f x x_1 h)) i) x) = (Ring.DirectLimit.of G' (fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f' x x_1 h)) i) ((g i) x)

@[simp]

theorem Ring.DirectLimit.map_id {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] : Ring.DirectLimit.map (fun (i : ι) => RingHom.id (G i)) ⋯ = RingHom.id (Ring.DirectLimit G fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f x x_1 h))

theorem Ring.DirectLimit.map_comp {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type v'} [(i : ι) → CommRing (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+* G' j} {G'' : ι → Type v''} [(i : ι) → CommRing (G'' i)] {f'' : (i j : ι) → i ≤ j → G'' i →+* G'' j} [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (g₁ : (i : ι) → G i →+* G' i) (g₂ : (i : ι) → G' i →+* G'' i) (hg₁ : ∀ (i j : ι) (h : i ≤ j), RingHom.comp (g₁ j) (f i j h) = RingHom.comp (f' i j h) (g₁ i)) (hg₂ : ∀ (i j : ι) (h : i ≤ j), RingHom.comp (g₂ j) (f' i j h) = RingHom.comp (f'' i j h) (g₂ i)) : RingHom.comp (Ring.DirectLimit.map g₂ hg₂) (Ring.DirectLimit.map g₁ hg₁) = Ring.DirectLimit.map (fun (i : ι) => RingHom.comp (g₂ i) (g₁ i)) ⋯

noncomputable def Ring.DirectLimit.congr {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type v'} [(i : ι) → CommRing (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+* G' j} [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (e : (i : ι) → G i ≃+* G' i) (he : ∀ (i j : ι) (h : i ≤ j), RingHom.comp (RingEquiv.toRingHom (e j)) (f i j h) = RingHom.comp (f' i j h) ↑(e i)) : (Ring.DirectLimit G fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f x x_1 h)) ≃+* Ring.DirectLimit G' fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f' x x_1 h)

Consider direct limits lim G and lim G' with direct system f and f' respectively, any family of equivalences eᵢ : Gᵢ ≅ G'ᵢ such that e ∘ f = f' ∘ e induces an equivalence lim G ⟶ lim G'.

Equations

• Ring.DirectLimit.congr e he = RingEquiv.ofHomInv (Ring.DirectLimit.map (fun (x : ι) => ↑(e x)) he) (Ring.DirectLimit.map (fun (i : ι) => ↑(RingEquiv.symm (e i))) ⋯) ⋯ ⋯

theorem Ring.DirectLimit.congr_apply_of {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type v'} [(i : ι) → CommRing (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+* G' j} [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (e : (i : ι) → G i ≃+* G' i) (he : ∀ (i j : ι) (h : i ≤ j), RingHom.comp (RingEquiv.toRingHom (e j)) (f i j h) = RingHom.comp (f' i j h) ↑(e i)) {i : ι} (g : G i) : (Ring.DirectLimit.congr e he) ((Ring.DirectLimit.of G (fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f x x_1 h)) i) g) = (Ring.DirectLimit.of G' (fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f' x x_1 h)) i) ((e i) g)

theorem Ring.DirectLimit.congr_symm_apply_of {ι : Type v} [Preorder ι] {G : ι → Type w} [(i : ι) → CommRing (G i)] {f : (i j : ι) → i ≤ j → G i →+* G j} {G' : ι → Type v'} [(i : ι) → CommRing (G' i)] {f' : (i j : ι) → i ≤ j → G' i →+* G' j} [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] (e : (i : ι) → G i ≃+* G' i) (he : ∀ (i j : ι) (h : i ≤ j), RingHom.comp (RingEquiv.toRingHom (e j)) (f i j h) = RingHom.comp (f' i j h) ↑(e i)) {i : ι} (g : G' i) : (RingEquiv.symm (Ring.DirectLimit.congr e he)) ((Ring.DirectLimit.of G' (fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f' x x_1 h)) i) g) = (Ring.DirectLimit.of G (fun (x x_1 : ι) (h : x ≤ x_1) => ⇑(f x x_1 h)) i) ((RingEquiv.symm (e i)) g)

noncomputable instance Field.DirectLimit.nontrivial {ι : Type v} [Preorder ι] (G : ι → Type w) [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] [(i : ι) → Field (G i)] (f' : (i j : ι) → i ≤ j → G i →+* G j) [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)] : Nontrivial (Ring.DirectLimit G fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h))

Equations

• ⋯ = ⋯

theorem Field.DirectLimit.exists_inv {ι : Type v} [Preorder ι] (G : ι → Type w) [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] [(i : ι) → Field (G i)] (f : (i j : ι) → i ≤ j → G i → G j) {p : Ring.DirectLimit G f} : p ≠ 0 → ∃ (y : Ring.DirectLimit G f), p * y = 1

noncomputable def Field.DirectLimit.inv {ι : Type v} [Preorder ι] (G : ι → Type w) [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] [(i : ι) → Field (G i)] (f : (i j : ι) → i ≤ j → G i → G j) (p : Ring.DirectLimit G f) : Ring.DirectLimit G f

Noncomputable multiplicative inverse in a direct limit of fields.

Equations

• Field.DirectLimit.inv G f p = if H : p = 0 then 0 else Classical.choose ⋯

theorem Field.DirectLimit.mul_inv_cancel {ι : Type v} [Preorder ι] (G : ι → Type w) [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] [(i : ι) → Field (G i)] (f : (i j : ι) → i ≤ j → G i → G j) {p : Ring.DirectLimit G f} (hp : p ≠ 0) : p * Field.DirectLimit.inv G f p = 1

theorem Field.DirectLimit.inv_mul_cancel {ι : Type v} [Preorder ι] (G : ι → Type w) [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] [(i : ι) → Field (G i)] (f : (i j : ι) → i ≤ j → G i → G j) {p : Ring.DirectLimit G f} (hp : p ≠ 0) : Field.DirectLimit.inv G f p * p = 1

@[reducible]

noncomputable def Field.DirectLimit.field {ι : Type v} [Preorder ι] (G : ι → Type w) [Nonempty ι] [IsDirected ι fun (x x_1 : ι) => x ≤ x_1] [(i : ι) → Field (G i)] (f' : (i j : ι) → i ≤ j → G i →+* G j) [DirectedSystem G fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h)] : Field (Ring.DirectLimit G fun (i j : ι) (h : i ≤ j) => ⇑(f' i j h))

Noncomputable field structure on the direct limit of fields. See note [reducible non-instances].

Equations

• Field.DirectLimit.field G f' = Field.mk ⋯ zpowRec ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (qsmulRec fun (a : ℚ) => ↑a) ⋯