Perfect pairings of modules #

A perfect pairing of two (left) modules may be defined either as:

A bilinear map M × N → R such that the induced maps M → Dual R N and N → Dual R M are both bijective. It follows from this that both M and N are reflexive modules.
A linear equivalence N ≃ Dual R M for which M is reflexive. (It then follows that N is reflexive.)

In this file we provide a PerfectPairing definition corresponding to 1 above, together with logic to connect 1 and 2.

Main definitions #

structure PerfectPairing (R : Type u_1) (M : Type u_2) (N : Type u_3) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] :

Type (max (max u_1 u_2) u_3)

A perfect pairing of two (left) modules over a commutative ring.

toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Function.Bijective ⇑self.toLin
bijectiveRight : Function.Bijective ⇑self.toLin.flip

Instances For

source

def PerfectPairing.mkOfInjective {K : Type u_4} {V : Type u_5} {W : Type u_6} [Field K] [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W] [FiniteDimensional K V] (B : V →ₗ[K] W →ₗ[K] K) (h : Function.Injective ⇑B) (h' : Function.Injective ⇑B.flip) :

PerfectPairing K V W

If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient to check only injectivity instead of bijectivity of the bilinear form.

Equations

PerfectPairing.mkOfInjective B h h' = { toLin := B, bijectiveLeft := ⋯, bijectiveRight := ⋯ }

Instances For

source

def PerfectPairing.mkOfInjective' {K : Type u_4} {V : Type u_5} {W : Type u_6} [Field K] [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W] [FiniteDimensional K W] (B : V →ₗ[K] W →ₗ[K] K) (h : Function.Injective ⇑B) (h' : Function.Injective ⇑B.flip) :

PerfectPairing K V W

If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient to check only injectivity instead of bijectivity of the bilinear form.

Equations

PerfectPairing.mkOfInjective' B h h' = { toLin := B, bijectiveLeft := ⋯, bijectiveRight := ⋯ }

Instances For

source

instance PerfectPairing.instFunLike {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] :

FunLike (PerfectPairing R M N) M (N →ₗ[R] R)

Equations

PerfectPairing.instFunLike = { coe := fun (f : PerfectPairing R M N) => ⇑f.toLin, coe_injective' := ⋯ }

source

@[simp]

theorem PerfectPairing.toLin_apply {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) {x : M} :

p.toLin x = p x

source

def PerfectPairing.flip {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) :

PerfectPairing R N M

Given a perfect pairing between M and N, we may interchange the roles of M and N.

Equations

p.flip = { toLin := p.toLin.flip, bijectiveLeft := ⋯, bijectiveRight := ⋯ }

Instances For

source

@[simp]

theorem PerfectPairing.flip_apply_apply {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) {x : M} {y : N} :

(p.flip y) x = (p x) y

source

@[simp]

theorem PerfectPairing.flip_flip {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) :

p.flip.flip = p

source

def PerfectPairing.toDualLeft {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) :

M ≃ₗ[R] Module.Dual R N

The linear equivalence from M to Dual R N induced by a perfect pairing.

Equations

p.toDualLeft = LinearEquiv.ofBijective p.toLin ⋯

Instances For

source

@[simp]

theorem PerfectPairing.toDualLeft_apply {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) (a : M) :

p.toDualLeft a = p a

source

@[simp]

theorem PerfectPairing.apply_toDualLeft_symm_apply {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) (f : Module.Dual R N) (x : N) :

(p (p.toDualLeft.symm f)) x = f x

source

def PerfectPairing.toDualRight {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) :

N ≃ₗ[R] Module.Dual R M

The linear equivalence from N to Dual R M induced by a perfect pairing.

Equations

p.toDualRight = p.flip.toDualLeft

Instances For

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@[simp]

theorem PerfectPairing.toDualRight_apply {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) (a : N) :

p.toDualRight a = p.flip a

source

@[simp]

theorem PerfectPairing.apply_apply_toDualRight_symm {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) (x : M) (f : Module.Dual R M) :

(p x) (p.toDualRight.symm f) = f x

source

theorem PerfectPairing.toDualLeft_of_toDualRight_symm {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) (x : M) (f : Module.Dual R M) :

(p.toDualLeft x) (p.toDualRight.symm f) = f x

source

theorem PerfectPairing.toDualRight_symm_toDualLeft {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) (x : M) :

p.toDualRight.symm.dualMap (p.toDualLeft x) = (Module.Dual.eval R M) x

source

theorem PerfectPairing.toDualRight_symm_comp_toDualLeft {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) :

↑p.toDualRight.symm.dualMap ∘ₗ ↑p.toDualLeft = Module.Dual.eval R M

source

theorem PerfectPairing.bijective_toDualRight_symm_toDualLeft {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) :

Function.Bijective fun (x : M) => p.toDualRight.symm.dualMap (p.toDualLeft x)

source

theorem PerfectPairing.reflexive_left {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) :

Module.IsReflexive R M

source

theorem PerfectPairing.reflexive_right {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) :

Module.IsReflexive R N

source

def PerfectPairing.restrict {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) {M' : Type u_4} {N' : Type u_5} [AddCommGroup M'] [Module R M'] [AddCommGroup N'] [Module R N'] (i : M' →ₗ[R] M) (j : N' →ₗ[R] N) (hM : IsCompl (LinearMap.range i) (Submodule.map p.toDualLeft.symm (LinearMap.range j).dualAnnihilator)) (hN : IsCompl (LinearMap.range j) (Submodule.map p.toDualRight.symm (LinearMap.range i).dualAnnihilator)) (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) :

PerfectPairing R M' N'

The restriction of a perfect pairing to submodules (expressed as injections to provide definitional control).

Equations

p.restrict i j hM hN hi hj = { toLin := p.toLin.compl₁₂ i j, bijectiveLeft := ⋯, bijectiveRight := ⋯ }

Instances For

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@[simp]

theorem PerfectPairing.restrict_toLin {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) {M' : Type u_4} {N' : Type u_5} [AddCommGroup M'] [Module R M'] [AddCommGroup N'] [Module R N'] (i : M' →ₗ[R] M) (j : N' →ₗ[R] N) (hM : IsCompl (LinearMap.range i) (Submodule.map p.toDualLeft.symm (LinearMap.range j).dualAnnihilator)) (hN : IsCompl (LinearMap.range j) (Submodule.map p.toDualRight.symm (LinearMap.range i).dualAnnihilator)) (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) :

(p.restrict i j hM hN hi hj).toLin = p.toLin.compl₁₂ i j

source

def PerfectPairing.restrictScalars {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) {S : Type u_4} [CommRing S] [Algebra S R] [Module S M] [Module S N] [IsScalarTower S R M] [IsScalarTower S R N] [NoZeroSMulDivisors S R] [Nontrivial R] {M' : Type u_5} {N' : Type u_6} [AddCommGroup M'] [Module S M'] [AddCommGroup N'] [Module S N'] (i : M' →ₗ[S] M) (j : N' →ₗ[S] N) (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) (hM : Submodule.span R ↑(LinearMap.range i) = ⊤) (hN : Submodule.span R ↑(LinearMap.range j) = ⊤) (h₁ : ∀ (g : Module.Dual S N'), ∃ (m : M'), ↑S (p.toDualLeft (i m)) ∘ₗ j = Algebra.linearMap S R ∘ₗ g) (h₂ : ∀ (g : Module.Dual S M'), ∃ (n : N'), ↑S (p.toDualRight (j n)) ∘ₗ i = Algebra.linearMap S R ∘ₗ g) (hp : ∀ (m : M') (n : N'), (p (i m)) (j n) ∈ (algebraMap S R).range) :

PerfectPairing S M' N'

Restriction of scalars for a perfect pairing taking values in a subring.

Equations

p.restrictScalars i j hi hj hM hN h₁ h₂ hp = { toLin := PerfectPairing.restrictScalarsAux p i j hp, bijectiveLeft := ⋯, bijectiveRight := ⋯ }

Instances For

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def PerfectPairing.restrictScalarsField {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : PerfectPairing R M N) [Nontrivial R] {M' : Type u_5} {N' : Type u_6} [AddCommGroup M'] [AddCommGroup N'] {K : Type u_7} [Field K] [Algebra K R] [Module K M] [Module K N] [IsScalarTower K R M] [IsScalarTower K R N] [Module K M'] [Module K N'] [FiniteDimensional K M'] (i : M' →ₗ[K] M) (j : N' →ₗ[K] N) (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) (hM : Submodule.span R ↑(LinearMap.range i) = ⊤) (hN : Submodule.span R ↑(LinearMap.range j) = ⊤) (hp : ∀ (m : M') (n : N'), (p (i m)) (j n) ∈ (algebraMap K R).range) :

PerfectPairing K M' N'

Restriction of scalars for a perfect pairing taking values in a subfield.

Equations

p.restrictScalarsField i j hi hj hM hN hp = PerfectPairing.mkOfInjective (PerfectPairing.restrictScalarsAux p i j hp) ⋯ ⋯

Instances For

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def IsReflexive.toPerfectPairingDual {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [Module.IsReflexive R M] :

PerfectPairing R (Module.Dual R M) M

A reflexive module has a perfect pairing with its dual.

Equations

IsReflexive.toPerfectPairingDual = { toLin := LinearMap.id, bijectiveLeft := ⋯, bijectiveRight := ⋯ }

Instances For

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@[simp]

theorem IsReflexive.toPerfectPairingDual_toLin {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [Module.IsReflexive R M] :

IsReflexive.toPerfectPairingDual.toLin = LinearMap.id

source

@[simp]

theorem IsReflexive.toPerfectPairingDual_apply {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [Module.IsReflexive R M] {f : Module.Dual R M} {x : M} :

(IsReflexive.toPerfectPairingDual f) x = f x

source

def LinearEquiv.flip {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) :

M ≃ₗ[R] Module.Dual R N

For a reflexive module M, an equivalence N ≃ₗ[R] Dual R M naturally yields an equivalence M ≃ₗ[R] Dual R N. Such equivalences are known as perfect pairings.

Equations

e.flip = (Module.evalEquiv R M).trans e.dualMap

Instances For

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@[simp]

theorem LinearEquiv.coe_toLinearMap_flip {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) :

↑e.flip = (↑e).flip

source

@[simp]

theorem LinearEquiv.flip_apply {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) (m : M) (n : N) :

(e.flip m) n = (e n) m

source

theorem LinearEquiv.symm_flip {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) :

e.flip.symm = e.symm.dualMap ≪≫ₗ (Module.evalEquiv R M).symm

source

theorem LinearEquiv.trans_dualMap_symm_flip {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) :

↑(e ≪≫ₗ e.flip.symm.dualMap) = Module.Dual.eval R N

source

theorem LinearEquiv.isReflexive_of_equiv_dual_of_isReflexive {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) :

Module.IsReflexive R N

If N is in perfect pairing with M, then it is reflexive.

source

@[simp]

theorem LinearEquiv.flip_flip {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) (h : Module.IsReflexive R N := ⋯) :

e.flip.flip = e

source

def LinearEquiv.toPerfectPairing {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) :

PerfectPairing R N M

If M is reflexive then a linear equivalence N ≃ Dual R M is a perfect pairing.

Equations

e.toPerfectPairing = { toLin := ↑e, bijectiveLeft := ⋯, bijectiveRight := ⋯ }

Instances For

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@[simp]

theorem LinearEquiv.toPerfectPairing_toLin {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) :

e.toPerfectPairing.toLin = ↑e

source

@[simp]

theorem Submodule.dualCoannihilator_map_linearEquiv_flip {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) (p : Submodule R M) :

(Submodule.map e.flip p).dualCoannihilator = Submodule.map e.symm p.dualAnnihilator

source

@[simp]

theorem Submodule.map_dualAnnihilator_linearEquiv_flip_symm {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) (p : Submodule R N) :

Submodule.map e.flip.symm p.dualAnnihilator = (Submodule.map e p).dualCoannihilator

source

@[simp]

theorem Submodule.map_dualCoannihilator_linearEquiv_flip {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) (p : Submodule R (Module.Dual R M)) :

Submodule.map e.flip p.dualCoannihilator = (Submodule.map e.symm p).dualAnnihilator

source

@[simp]

theorem Submodule.dualAnnihilator_map_linearEquiv_flip_symm {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.IsReflexive R M] (e : N ≃ₗ[R] Module.Dual R M) (p : Submodule R (Module.Dual R N)) :

(Submodule.map e.flip.symm p).dualAnnihilator = Submodule.map e p.dualCoannihilator

Documentation

Mathlib.LinearAlgebra.PerfectPairing

Perfect pairings of modules #

Main definitions #