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

• 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 (_ : i ≤ k) x

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

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 h => 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 h => 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 (_ : 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 h => ↑(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 h => ↑(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 (_ : 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.

def Module.DirectLimit {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] [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) : Type (max v w)

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

Equations

• Module.DirectLimit G f = (DirectSum ι G ⧸ Submodule.span R {a | ∃ i j H x, ↑(DirectSum.lof R ι G i) x - ↑(DirectSum.lof R ι G j) (↑(f i j H) x) = a})

instance Module.DirectLimit.addCommGroup {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] [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) : AddCommGroup (Module.DirectLimit G f)

Equations

• Module.DirectLimit.addCommGroup G f = Submodule.Quotient.addCommGroup (Submodule.span R {a | ∃ i j H x, ↑(DirectSum.lof R ι G i) x - ↑(DirectSum.lof R ι G j) (↑(f i j H) x) = a})

instance Module.DirectLimit.module {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] [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) : Module R (Module.DirectLimit G f)

Equations

• Module.DirectLimit.module G f = Submodule.Quotient.module (Submodule.span R {a | ∃ i j H x, ↑(DirectSum.lof R ι G i) x - ↑(DirectSum.lof R ι G j) (↑(f i j H) x) = a})

instance Module.DirectLimit.inhabited {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] [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) : Inhabited (Module.DirectLimit G f)

Equations

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

def Module.DirectLimit.of (R : Type u) [Ring R] (ι : Type v) [dec_ι : DecidableEq ι] [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) : 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} [dec_ι : DecidableEq ι] [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} {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} [dec_ι : DecidableEq ι] [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} [Nonempty ι] [IsDirected ι fun x x_1 => x ≤ x_1] (z : Module.DirectLimit G f) : ∃ i x, ↑(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} [dec_ι : DecidableEq ι] [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} [Nonempty ι] [IsDirected ι fun x x_1 => 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

def Module.DirectLimit.lift (R : Type u) [Ring R] (ι : Type v) [dec_ι : DecidableEq ι] [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) {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} [dec_ι : DecidableEq ι] [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} {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} [dec_ι : DecidableEq ι] [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} {P : Type u₁} [AddCommGroup P] [Module R P] [Nonempty ι] [IsDirected ι fun x x_1 => 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 => LinearMap.comp F (Module.DirectLimit.of R ι G f i)) (_ : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑((fun i => LinearMap.comp F (Module.DirectLimit.of R ι G f i)) j) (↑(f i j hij) x) = ↑((fun i => LinearMap.comp F (Module.DirectLimit.of R ι G f i)) i) x)) x

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} [dec_ι : DecidableEq ι] [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 h => ↑(f i j h)] {x : DirectSum ι G} {i : ι} {j : ι} (hij : i ≤ j) (hx : ∀ (k : ι), k ∈ DFinsupp.support x → k ≤ i) : ↑(DirectSum.toModule R ι (G j) fun k => Module.DirectLimit.totalize G f k j) x = ↑(f i j hij) (↑(DirectSum.toModule R ι (G i) fun k => Module.DirectLimit.totalize G f k i) x)

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

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

def AddCommGroup.DirectLimit {ι : Type v} [dec_ι : DecidableEq ι] [Preorder ι] (G : ι → Type w) [(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 hij => 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 h => ↑(f i j h)] : DirectedSystem G fun i j hij => ↑(AddMonoidHom.toIntLinearMap (f i j hij))

instance AddCommGroup.DirectLimit.instAddCommGroupDirectLimit {ι : Type v} [dec_ι : DecidableEq ι] [Preorder ι] (G : ι → Type w) [(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 hij => AddMonoidHom.toIntLinearMap (f i j hij)

instance AddCommGroup.DirectLimit.instInhabitedDirectLimit {ι : Type v} [dec_ι : DecidableEq ι] [Preorder ι] (G : ι → Type w) [(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 }

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

The canonical map from a component to the direct limit.

Equations

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

@[simp]

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

def AddCommGroup.DirectLimit.lift {ι : Type v} [dec_ι : DecidableEq ι] [Preorder ι] (G : ι → Type w) [(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 →ₗ[ℤ] 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

• AddCommGroup.DirectLimit.lift G f P g Hg = Module.DirectLimit.lift ℤ ι G (fun i j hij => AddMonoidHom.toIntLinearMap (f i j hij)) (fun i => AddMonoidHom.toIntLinearMap (g i)) Hg

@[simp]

theorem AddCommGroup.DirectLimit.lift_of {ι : Type v} [dec_ι : DecidableEq ι] [Preorder ι] {G : ι → Type w} [(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} [dec_ι : DecidableEq ι] [Preorder ι] {G : ι → Type w} [(i : ι) → AddCommGroup (G i)] {f : (i j : ι) → i ≤ j → G i →+ G j} (P : Type u₁) [AddCommGroup P] [Nonempty ι] [IsDirected ι fun x x_1 => x ≤ x_1] (F : AddCommGroup.DirectLimit G f →+ P) (x : AddCommGroup.DirectLimit G f) : ↑F x = ↑(AddCommGroup.DirectLimit.lift G f P (fun i => AddMonoidHom.comp F (LinearMap.toAddMonoidHom (AddCommGroup.DirectLimit.of G f i))) (_ : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑F (↑(AddCommGroup.DirectLimit.of G f j) (↑(f i j hij) x)) = ↑F (↑(AddCommGroup.DirectLimit.of G f i) x))) x

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.

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.

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

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 }

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 }

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 => x ≤ x_1] (z : Ring.DirectLimit G f) : ∃ i x, ↑(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 => x ≤ x_1] (q : Polynomial (Ring.DirectLimit G fun i j h => ↑(f' i j h))) : ∃ i p, Polynomial.map (Ring.DirectLimit.of G (fun i j h => ↑(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 => 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 h => ↑(f' i j h)] {x : FreeCommRing ((i : ι) × G i)} {s : Set ((i : ι) × G i)} {t : Set ((i : ι) × G i)} (hxs : FreeCommRing.IsSupported x s) {j : ι} {k : ι} (hj : ∀ (z : (i : ι) × G i), z ∈ s → z.fst ≤ j) (hk : ∀ (z : (i : ι) × G i), z ∈ t → z.fst ≤ k) (hjk : j ≤ k) (hst : s ⊆ t) : ↑(f' j k hjk) (↑(↑FreeCommRing.lift fun ix => ↑(f' (↑ix).fst j (hj ↑ix (_ : ↑ix ∈ s))) (↑ix).snd) (↑(FreeCommRing.restriction s) x)) = ↑(↑FreeCommRing.lift fun ix => ↑(f' (↑ix).fst k (hk ↑ix (_ : ↑ix ∈ t))) (↑ix).snd) (↑(FreeCommRing.restriction t) x)

theorem Ring.DirectLimit.of.zero_exact_aux {ι : Type v} [dec_ι : DecidableEq ι] [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 h => ↑(f' i j h)] [Nonempty ι] [IsDirected ι fun x x_1 => x ≤ x_1] {x : FreeCommRing ((i : ι) × G i)} (H : ↑(Ideal.Quotient.mk (Ideal.span {a | (∃ i j H x, FreeCommRing.of { fst := j, snd := (fun i j h => ↑(f' i j h)) i j H x } - FreeCommRing.of { fst := i, snd := x } = a) ∨ (∃ i, FreeCommRing.of { fst := i, snd := 1 } - 1 = a) ∨ (∃ i x y, FreeCommRing.of { fst := i, snd := x + y } - (FreeCommRing.of { fst := i, snd := x } + FreeCommRing.of { fst := i, snd := y }) = a) ∨ ∃ i x y, FreeCommRing.of { fst := i, snd := x * y } - FreeCommRing.of { fst := i, snd := x } * FreeCommRing.of { fst := i, snd := y } = a})) x = 0) : ∃ j s H, FreeCommRing.IsSupported x s ∧ ↑(↑FreeCommRing.lift fun ix => ↑(f' (↑ix).fst j (_ : (↑ix).fst ≤ j)) (↑ix).snd) (↑(FreeCommRing.restriction s) x) = 0

theorem Ring.DirectLimit.of.zero_exact {ι : Type v} [dec_ι : DecidableEq ι] [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 h => ↑(f' i j h)] [IsDirected ι fun x x_1 => x ≤ x_1] {i : ι} {x : G i} (hix : ↑(Ring.DirectLimit.of G (fun i j h => ↑(f' i j h)) i) x = 0) : ∃ j hij, ↑(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} [dec_ι : DecidableEq ι] [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 => x ≤ x_1] [DirectedSystem G fun i j h => ↑(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 h => ↑(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.

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] [Nonempty ι] [IsDirected ι fun x x_1 => x ≤ x_1] (F : Ring.DirectLimit G f →+* P) (x : Ring.DirectLimit G f) : ↑F x = ↑(Ring.DirectLimit.lift G f P (fun i => RingHom.comp F (Ring.DirectLimit.of G f i)) (_ : ∀ (i j : ι) (hij : i ≤ j) (x : G i), ↑F (↑(Ring.DirectLimit.of G f j) (f i j hij x)) = ↑F (↑(Ring.DirectLimit.of G f i) x))) x

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

Equations

• @Field.DirectLimit.nontrivial = @Field.DirectLimit.nontrivial.proof_1

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

theorem Field.DirectLimit.mul_inv_cancel {ι : Type v} [Preorder ι] (G : ι → Type w) [Nonempty ι] [IsDirected ι fun x x_1 => 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 => 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} [dec_ι : DecidableEq ι] [Preorder ι] (G : ι → Type w) [Nonempty ι] [IsDirected ι fun x x_1 => x ≤ x_1] [(i : ι) → Field (G i)] (f' : (i j : ι) → i ≤ j → G i →+* G j) [DirectedSystem G fun i j h => ↑(f' i j h)] : Field (Ring.DirectLimit G fun i j h => ↑(f' i j h))

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

Equations

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