Documentation

Mathlib.LinearAlgebra.ExteriorAlgebra.Basic

Exterior Algebras #

We construct the exterior algebra of a module M over a commutative semiring R.

Notation #

The exterior algebra of the R-module M is denoted as ExteriorAlgebra R M. It is endowed with the structure of an R-algebra.

The nth exterior power of the R-module M is denoted by exteriorPower R n M; it is of type Submodule R (ExteriorAlgebra R M) and defined as LinearMap.range (ExteriorAlgebra.ι R : M →ₗ[R] ExteriorAlgebra R M) ^ n. We also introduce the notation ⋀[R]^n M for exteriorPower R n M.

Given a linear morphism f : M → A from a module M to another R-algebra A, such that cond : ∀ m : M, f m * f m = 0, there is a (unique) lift of f to an R-algebra morphism, which is denoted ExteriorAlgebra.lift R f cond.

The canonical linear map M → ExteriorAlgebra R M is denoted ExteriorAlgebra.ι R.

Theorems #

The main theorems proved ensure that ExteriorAlgebra R M satisfies the universal property of the exterior algebra.

  1. ι_comp_lift is the fact that the composition of ι R with lift R f cond agrees with f.
  2. lift_unique ensures the uniqueness of lift R f cond with respect to 1.

Definitions #

Implementation details #

The exterior algebra of M is constructed as simply CliffordAlgebra (0 : QuadraticForm R M), as this avoids us having to duplicate API.

@[reducible, inline]
abbrev ExteriorAlgebra (R : Type u1) [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] :
Type (max u2 u1)

The exterior algebra of an R-module M.

Equations
@[reducible, inline]
abbrev ExteriorAlgebra.ι (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] :

The canonical linear map M →ₗ[R] ExteriorAlgebra R M.

Equations
@[reducible, inline]
abbrev ExteriorAlgebra.exteriorPower (R : Type u1) [CommRing R] (n : ) (M : Type u2) [AddCommGroup M] [Module R M] :

Definition of the nth exterior power of a R-module N. We introduce the notation ⋀[R]^n M for exteriorPower R n M.

Equations

Definition of the nth exterior power of a R-module N. We introduce the notation ⋀[R]^n M for exteriorPower R n M.

Equations
  • One or more equations did not get rendered due to their size.
theorem ExteriorAlgebra.ι_sq_zero {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (m : M) :
(ι R) m * (ι R) m = 0

As well as being linear, ι m squares to zero.

theorem ExteriorAlgebra.comp_ι_sq_zero {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (g : ExteriorAlgebra R M →ₐ[R] A) (m : M) :
g ((ι R) m) * g ((ι R) m) = 0
def ExteriorAlgebra.lift (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] :
{ f : M →ₗ[R] A // ∀ (m : M), f m * f m = 0 } (ExteriorAlgebra R M →ₐ[R] A)

Given a linear map f : M →ₗ[R] A into an R-algebra A, which satisfies the condition: cond : ∀ m : M, f m * f m = 0, this is the canonical lift of f to a morphism of R-algebras from ExteriorAlgebra R M to A.

Equations
@[simp]
theorem ExteriorAlgebra.lift_symm_apply (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (a✝ : ExteriorAlgebra R M →ₐ[R] A) :
(lift R).symm a✝ = ((CliffordAlgebra.lift 0).symm a✝),
@[simp]
theorem ExteriorAlgebra.ι_comp_lift (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = 0) :
((lift R) f, cond).toLinearMap ∘ₗ ι R = f
@[simp]
theorem ExteriorAlgebra.lift_ι_apply (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = 0) (x : M) :
((lift R) f, cond) ((ι R) x) = f x
theorem ExteriorAlgebra.lift_unique (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = 0) (g : ExteriorAlgebra R M →ₐ[R] A) :
g.toLinearMap ∘ₗ ι R = f g = (lift R) f, cond
@[simp]
theorem ExteriorAlgebra.lift_comp_ι {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] (g : ExteriorAlgebra R M →ₐ[R] A) :
theorem ExteriorAlgebra.hom_ext {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] {f g : ExteriorAlgebra R M →ₐ[R] A} (h : f.toLinearMap ∘ₗ ι R = g.toLinearMap ∘ₗ ι R) :
f = g

See note [partially-applied ext lemmas].

theorem ExteriorAlgebra.hom_ext_iff {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {A : Type u_1} [Semiring A] [Algebra R A] {f g : ExteriorAlgebra R M →ₐ[R] A} :
theorem ExteriorAlgebra.induction {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {C : ExteriorAlgebra R MProp} (algebraMap : ∀ (r : R), C ((algebraMap R (ExteriorAlgebra R M)) r)) (ι : ∀ (x : M), C ((ι R) x)) (mul : ∀ (a b : ExteriorAlgebra R M), C aC bC (a * b)) (add : ∀ (a b : ExteriorAlgebra R M), C aC bC (a + b)) (a : ExteriorAlgebra R M) :
C a

If C holds for the algebraMap of r : R into ExteriorAlgebra R M, the ι of x : M, and is preserved under addition and multiplication, then it holds for all of ExteriorAlgebra R M.

@[simp]
theorem ExteriorAlgebra.algebraMap_inj {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (x y : R) :
@[simp]
theorem ExteriorAlgebra.algebraMap_eq_zero_iff {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (x : R) :
(algebraMap R (ExteriorAlgebra R M)) x = 0 x = 0
@[simp]
theorem ExteriorAlgebra.algebraMap_eq_one_iff {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (x : R) :
(algebraMap R (ExteriorAlgebra R M)) x = 1 x = 1

Invertibility in the exterior algebra is the same as invertibility of the base ring.

Equations

The left-inverse of ι.

As an implementation detail, we implement this using TrivSqZeroExt which has a suitable algebra structure.

Equations
@[simp]
theorem ExteriorAlgebra.ι_inj (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (x y : M) :
(ι R) x = (ι R) y x = y
@[simp]
theorem ExteriorAlgebra.ι_eq_zero_iff {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (x : M) :
(ι R) x = 0 x = 0
@[simp]
theorem ExteriorAlgebra.ι_eq_algebraMap_iff {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (x : M) (r : R) :
(ι R) x = (algebraMap R (ExteriorAlgebra R M)) r x = 0 r = 0
@[simp]
theorem ExteriorAlgebra.ι_ne_one {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] [Nontrivial R] (x : M) :
(ι R) x 1

The generators of the exterior algebra are disjoint from its scalars.

@[simp]
theorem ExteriorAlgebra.ι_add_mul_swap {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (x y : M) :
(ι R) x * (ι R) y + (ι R) y * (ι R) x = 0
theorem ExteriorAlgebra.ι_mul_prod_list {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {n : } (f : Fin nM) (i : Fin n) :
(ι R) (f i) * (List.ofFn fun (i : Fin n) => (ι R) (f i)).prod = 0

The product of n terms of the form ι R m is an alternating map.

This is a special case of MultilinearMap.mkPiAlgebraFin, and the exterior algebra version of TensorAlgebra.tprod.

Equations
  • One or more equations did not get rendered due to their size.
theorem ExteriorAlgebra.ιMulti_apply {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {n : } (v : Fin nM) :
(ιMulti R n) v = (List.ofFn fun (i : Fin n) => (ι R) (v i)).prod
@[simp]
theorem ExteriorAlgebra.ιMulti_zero_apply {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (v : Fin 0M) :
(ιMulti R 0) v = 1
@[simp]
theorem ExteriorAlgebra.ιMulti_succ_apply {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {n : } (v : Fin n.succM) :
(ιMulti R n.succ) v = (ι R) (v 0) * (ιMulti R n) (Matrix.vecTail v)
theorem ExteriorAlgebra.ιMulti_range (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (n : ) :
Set.range (ιMulti R n) (⋀[R]^n M)

The image of ExteriorAlgebra.ιMulti R n is contained in the nth exterior power.

The image of ExteriorAlgebra.ιMulti R n spans the nth exterior power, as a submodule of the exterior algebra.

@[reducible, inline]
abbrev ExteriorAlgebra.ιMulti_family (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (n : ) {I : Type u_1} [LinearOrder I] (v : IM) (s : { s : Finset I // s.card = n }) :

Given a linearly ordered family v of vectors of M and a natural number n, produce the family of nfold exterior products of elements of v, seen as members of the exterior algebra.

Equations

An ExteriorAlgebra over a nontrivial ring is nontrivial.

Functoriality of the exterior algebra.

def ExteriorAlgebra.map {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) :

The morphism of exterior algebras induced by a linear map.

Equations
@[simp]
theorem ExteriorAlgebra.map_comp_ι {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) :
@[simp]
theorem ExteriorAlgebra.map_apply_ι {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) (m : M) :
(map f) ((ι R) m) = (ι R) (f m)
@[simp]
theorem ExteriorAlgebra.map_apply_ιMulti {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {n : } (f : M →ₗ[R] N) (m : Fin nM) :
(map f) ((ιMulti R n) m) = (ιMulti R n) (f m)
@[simp]
theorem ExteriorAlgebra.map_comp_ιMulti {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {n : } (f : M →ₗ[R] N) :
@[simp]
theorem ExteriorAlgebra.map_comp_map {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} {N' : Type u5} [AddCommGroup N] [Module R N] [AddCommGroup N'] [Module R N'] (f : M →ₗ[R] N) (g : N →ₗ[R] N') :
(map g).comp (map f) = map (g ∘ₗ f)
theorem ExteriorAlgebra.ιInv_comp_map {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) :
@[simp]
theorem ExteriorAlgebra.leftInverse_map_iff {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} {g : N →ₗ[R] M} :

For a linear map f from M to N, ExteriorAlgebra.map g is a retraction of ExteriorAlgebra.map f iff g is a retraction of f.

theorem ExteriorAlgebra.map_injective {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} (hf : ∃ (g : N →ₗ[R] M), g ∘ₗ f = LinearMap.id) :

A morphism of modules that admits a linear retraction induces an injective morphism of exterior algebras.

@[simp]
theorem ExteriorAlgebra.map_surjective_iff {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} :

A morphism of modules is surjective if and only the morphism of exterior algebras that it induces is surjective.

theorem ExteriorAlgebra.map_injective_field {K : Type u_1} {E : Type u_2} {F : Type u_3} [Field K] [AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F] {f : E →ₗ[K] F} (hf : LinearMap.ker f = ) :

An injective morphism of vector spaces induces an injective morphism of exterior algebras.

@[simp]
theorem TensorAlgebra.toExterior_ι {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (m : M) :