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.

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

• ιMulti is the AlternatingMap corresponding to the wedge product of ι R m terms.

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]

def 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

• ExteriorAlgebra R M = CliffordAlgebra 0

@[reducible]

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

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

Equations

• ExteriorAlgebra.ι R = CliffordAlgebra.ι 0

theorem ExteriorAlgebra.ι_sq_zero {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (m : M) : (ExteriorAlgebra.ι R) m * (ExteriorAlgebra.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 ((ExteriorAlgebra.ι R) m) * g ((ExteriorAlgebra.ι R) m) = 0

@[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), (ExteriorAlgebra.lift R).symm a = { val := ↑((CliffordAlgebra.lift 0).symm a), property := (_ : ∀ (m : M), ↑((CliffordAlgebra.lift 0).symm a) m * ↑((CliffordAlgebra.lift 0).symm a) 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

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

@[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) : AlgHom.toLinearMap ((ExteriorAlgebra.lift R) { val := f, property := cond }) ∘ₗ ExteriorAlgebra.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) : ((ExteriorAlgebra.lift R) { val := f, property := cond }) ((ExteriorAlgebra.ι R) x) = f x

@[simp]

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) : AlgHom.toLinearMap g ∘ₗ ExteriorAlgebra.ι R = f ↔ g = (ExteriorAlgebra.lift R) { val := f, property := 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) : (ExteriorAlgebra.lift R) { val := AlgHom.toLinearMap g ∘ₗ ExteriorAlgebra.ι R, property := (_ : ∀ (m : M), g ((ExteriorAlgebra.ι R) m) * g ((ExteriorAlgebra.ι R) m) = 0) } = g

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 : ExteriorAlgebra R M →ₐ[R] A} {g : ExteriorAlgebra R M →ₐ[R] A} (h : AlgHom.toLinearMap f ∘ₗ ExteriorAlgebra.ι R = AlgHom.toLinearMap g ∘ₗ ExteriorAlgebra.ι R) : f = g

See note [partially-applied ext lemmas].

theorem ExteriorAlgebra.induction {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {C : ExteriorAlgebra R M → Prop} (h_grade0 : ∀ (r : R), C ((algebraMap R (ExteriorAlgebra R M)) r)) (h_grade1 : ∀ (x : M), C ((ExteriorAlgebra.ι R) x)) (h_mul : ∀ (a b : ExteriorAlgebra R M), C a → C b → C (a * b)) (h_add : ∀ (a b : ExteriorAlgebra R M), C a → C b → C (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 muliplication, then it holds for all of ExteriorAlgebra R M.

def ExteriorAlgebra.algebraMapInv {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] : ExteriorAlgebra R M →ₐ[R] R

The left-inverse of algebraMap.

Equations

• ExteriorAlgebra.algebraMapInv = (ExteriorAlgebra.lift R) { val := 0, property := (_ : M → 0 * 0 = 0) }

theorem ExteriorAlgebra.algebraMap_leftInverse {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] : Function.LeftInverse ⇑ExteriorAlgebra.algebraMapInv ⇑(algebraMap R (ExteriorAlgebra R M))

@[simp]

theorem ExteriorAlgebra.algebraMap_inj {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (x : R) (y : R) : (algebraMap R (ExteriorAlgebra R M)) x = (algebraMap R (ExteriorAlgebra R M)) y ↔ x = y

@[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

theorem ExteriorAlgebra.isUnit_algebraMap {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (r : R) : IsUnit ((algebraMap R (ExteriorAlgebra R M)) r) ↔ IsUnit r

@[simp]

theorem ExteriorAlgebra.invertibleAlgebraMapEquiv_symm_apply_invOf_toQuot {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (r : R) : ∀ (x : Invertible r), (⅟((algebraMap R (ExteriorAlgebra R M)) r)).toQuot = Quot.mk (RingQuot.Rel (CliffordAlgebra.Rel 0)) ((algebraMap R (TensorAlgebra R M)) ⅟r)

@[simp]

theorem ExteriorAlgebra.invertibleAlgebraMapEquiv_apply_invOf {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (r : R) : ∀ (x : Invertible ((algebraMap R (ExteriorAlgebra R M)) r)), ⅟r = ⅟(ExteriorAlgebra.algebraMapInv ((algebraMap R (ExteriorAlgebra R M)) r))

def ExteriorAlgebra.invertibleAlgebraMapEquiv {R : Type u1} [CommRing R] (M : Type u2) [AddCommGroup M] [Module R M] (r : R) : Invertible ((algebraMap R (ExteriorAlgebra R M)) r) ≃ Invertible r

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

Equations

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

def ExteriorAlgebra.toTrivSqZeroExt {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] : ExteriorAlgebra R M →ₐ[R] TrivSqZeroExt R M

The canonical map from ExteriorAlgebra R M into TrivSqZeroExt R M that sends ExteriorAlgebra.ι to TrivSqZeroExt.inr.

Equations

• ExteriorAlgebra.toTrivSqZeroExt = (ExteriorAlgebra.lift R) { val := TrivSqZeroExt.inrHom R M, property := (_ : ∀ (m : M), TrivSqZeroExt.inr m * TrivSqZeroExt.inr m = 0) }

@[simp]

theorem ExteriorAlgebra.toTrivSqZeroExt_ι {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] (x : M) : ExteriorAlgebra.toTrivSqZeroExt ((ExteriorAlgebra.ι R) x) = TrivSqZeroExt.inr x

def ExteriorAlgebra.ιInv {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] : ExteriorAlgebra R M →ₗ[R] M

The left-inverse of ι.

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

Equations

• ExteriorAlgebra.ιInv = LinearMap.comp (TrivSqZeroExt.sndHom R M) (AlgHom.toLinearMap ExteriorAlgebra.toTrivSqZeroExt)

theorem ExteriorAlgebra.ι_leftInverse {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] : Function.LeftInverse ⇑ExteriorAlgebra.ιInv.toAddHom ⇑(ExteriorAlgebra.ι R)

@[simp]

theorem ExteriorAlgebra.ι_inj (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (x : M) (y : M) : (ExteriorAlgebra.ι R) x = (ExteriorAlgebra.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) : (ExteriorAlgebra.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) : (ExteriorAlgebra.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) : (ExteriorAlgebra.ι R) x ≠ 1

theorem ExteriorAlgebra.ι_range_disjoint_one {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] : Disjoint (LinearMap.range (ExteriorAlgebra.ι R)) 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 : M) (y : M) : (ExteriorAlgebra.ι R) x * (ExteriorAlgebra.ι R) y + (ExteriorAlgebra.ι R) y * (ExteriorAlgebra.ι R) x = 0

theorem ExteriorAlgebra.ι_mul_prod_list {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {n : ℕ} (f : Fin n → M) (i : Fin n) : (ExteriorAlgebra.ι R) (f i) * List.prod (List.ofFn fun (i : Fin n) => (ExteriorAlgebra.ι R) (f i)) = 0

def ExteriorAlgebra.ιMulti (R : Type u1) [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (n : ℕ) : M [Λ^Fin n]→ₗ[R] ExteriorAlgebra R M

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 : ℕ} (v : Fin n → M) : (ExteriorAlgebra.ιMulti R n) v = List.prod (List.ofFn fun (i : Fin n) => (ExteriorAlgebra.ι R) (v i))

@[simp]

theorem ExteriorAlgebra.ιMulti_zero_apply {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (v : Fin 0 → M) : (ExteriorAlgebra.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 : ℕ} (v : Fin (Nat.succ n) → M) : (ExteriorAlgebra.ιMulti R (Nat.succ n)) v = (ExteriorAlgebra.ι R) (v 0) * (ExteriorAlgebra.ιMulti R n) (Matrix.vecTail v)

theorem ExteriorAlgebra.ιMulti_succ_curryLeft {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {n : ℕ} (m : M) : (AlternatingMap.curryLeft (ExteriorAlgebra.ιMulti R (Nat.succ n))) m = (LinearMap.compAlternatingMap (LinearMap.mulLeft R ((ExteriorAlgebra.ι R) m))) (ExteriorAlgebra.ιMulti R n)

instance ExteriorAlgebra.instNontrivialExteriorAlgebra {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] [Nontrivial R] : Nontrivial (ExteriorAlgebra R M)

An ExteriorAlgebra over a nontrivial ring is nontrivial.

Equations

• (_ : Nontrivial (ExteriorAlgebra R M)) = (_ : Nontrivial (ExteriorAlgebra R M))

def TensorAlgebra.toExterior {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] : TensorAlgebra R M →ₐ[R] ExteriorAlgebra R M

The canonical image of the TensorAlgebra in the ExteriorAlgebra, which maps TensorAlgebra.ι R x to ExteriorAlgebra.ι R x.

Equations

• TensorAlgebra.toExterior = (TensorAlgebra.lift R) (ExteriorAlgebra.ι R)

@[simp]

theorem TensorAlgebra.toExterior_ι {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (m : M) : TensorAlgebra.toExterior ((TensorAlgebra.ι R) m) = (ExteriorAlgebra.ι R) m