# mathlibdocumentation

linear_algebra.exterior_algebra

# 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 exterior_algebra 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 exterior_algebra.lift R f cond.

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

## Theorems #

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

1. ι_comp_lift is 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 #

• ι_multi is the alternating_map corresponding to the wedge product of ι R m terms.

## Implementation details #

The exterior algebra of M is constructed as a quotient of the tensor algebra, as follows.

1. We define a relation exterior_algebra.rel R M on tensor_algebra R M. This is the smallest relation which identifies squares of elements of M with 0.
2. The exterior algebra is the quotient of the tensor algebra by this relation.
inductive exterior_algebra.rel (R : Type u1) (M : Type u2) [ M] :
→ Prop
• of : ∀ (R : Type u1) [_inst_1 : (M : Type u2) [_inst_2 : [_inst_3 : M] (m : M), (( m) * m) 0

rel relates each ι m * ι m, for m : M, with 0.

The exterior algebra of M is defined as the quotient modulo this relation.

def exterior_algebra (R : Type u1) (M : Type u2) [ M] :
Type (max u1 u2)

The exterior algebra of an R-module M.

Equations
@[instance]
def exterior_algebra.inhabited (R : Type u1) (M : Type u2) [ M] :
@[instance]
def exterior_algebra.semiring (R : Type u1) (M : Type u2) [ M] :
@[instance]
def exterior_algebra.algebra (R : Type u1) (M : Type u2) [ M] :
M)
@[instance]
def exterior_algebra.ring {M : Type u2} {S : Type u3} [comm_ring S] [ M] :
ring M)
Equations
def exterior_algebra.ι (R : Type u1) {M : Type u2} [ M] :

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

Equations
@[simp]
theorem exterior_algebra.ι_sq_zero {R : Type u1} {M : Type u2} [ M] (m : M) :
( m) * m = 0

As well as being linear, ι m squares to zero

@[simp]
theorem exterior_algebra.comp_ι_sq_zero {R : Type u1} {M : Type u2} [ M] {A : Type u_1} [semiring A] [ A] (g : →ₐ[R] A) (m : M) :
(g ( m)) * g ( m) = 0
def exterior_algebra.lift (R : Type u1) {M : Type u2} [ M] {A : Type u_1} [semiring A] [ A] :
{f // ∀ (m : M), (f m) * f m = 0} 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 exterior_algebra R M to A.

Equations
@[simp]
theorem exterior_algebra.lift_symm_apply (R : Type u1) {M : Type u2} [ M] {A : Type u_1} [semiring A] [ A] (F : →ₐ[R] A) :
.symm) F = , _⟩
@[simp]
theorem exterior_algebra.ι_comp_lift (R : Type u1) {M : Type u2} [ M] {A : Type u_1} [semiring A] [ A] (f : M →ₗ[R] A) (cond : ∀ (m : M), (f m) * f m = 0) :
f, cond⟩).to_linear_map.comp = f
@[simp]
theorem exterior_algebra.lift_ι_apply (R : Type u1) {M : Type u2} [ M] {A : Type u_1} [semiring A] [ A] (f : M →ₗ[R] A) (cond : ∀ (m : M), (f m) * f m = 0) (x : M) :
f, cond⟩) ( x) = f x
@[simp]
theorem exterior_algebra.lift_unique (R : Type u1) {M : Type u2} [ M] {A : Type u_1} [semiring A] [ A] (f : M →ₗ[R] A) (cond : ∀ (m : M), (f m) * f m = 0) (g : →ₐ[R] A) :
= f g = f, cond⟩
@[simp]
theorem exterior_algebra.lift_comp_ι {R : Type u1} {M : Type u2} [ M] {A : Type u_1} [semiring A] [ A] (g : →ₐ[R] A) :
, _⟩ = g
@[ext]
theorem exterior_algebra.hom_ext {R : Type u1} {M : Type u2} [ M] {A : Type u_1} [semiring A] [ A] {f g : →ₐ[R] A} (h : = ) :
f = g
theorem exterior_algebra.induction {R : Type u1} {M : Type u2} [ M] {C : → Prop} (h_grade0 : ∀ (r : R), C ( M)) r)) (h_grade1 : ∀ (x : M), C ( x)) (h_mul : ∀ (a b : M), C aC bC (a * b)) (h_add : ∀ (a b : M), C aC bC (a + b)) (a : M) :
C a

If C holds for the algebra_map of r : R into exterior_algebra R M, the ι of x : M, and is preserved under addition and muliplication, then it holds for all of exterior_algebra R M.

def exterior_algebra.algebra_map_inv {R : Type u1} {M : Type u2} [ M] :

The left-inverse of algebra_map.

Equations
theorem exterior_algebra.algebra_map_left_inverse {R : Type u1} {M : Type u2} [ M] :
def exterior_algebra.ι_inv {R : Type u1} {M : Type u2} [ M] :

The left-inverse of ι.

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

Equations
theorem exterior_algebra.ι_left_inverse {R : Type u1} {M : Type u2} [ M] :
@[simp]
theorem exterior_algebra.ι_add_mul_swap {R : Type u1} {M : Type u2} [ M] (x y : M) :
( x) * y + ( y) * x = 0
theorem exterior_algebra.ι_mul_prod_list {R : Type u1} {M : Type u2} [ M] {n : } (f : fin n → M) (i : fin n) :
( (f i)) * (list.of_fn (λ (i : fin n), (f i))).prod = 0
def exterior_algebra.ι_multi (R : Type u1) {M : Type u2} [ M] (n : ) :
M) (fin n)

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

This is a special case of multilinear_map.mk_pi_algebra_fin

Equations
theorem exterior_algebra.ι_multi_apply {R : Type u1} {M : Type u2} [ M] {n : } (v : fin n → M) :
v = (list.of_fn (λ (i : fin n), (v i))).prod
def tensor_algebra.to_exterior {R : Type u1} {M : Type u2} [ M] :

The canonical image of the tensor_algebra in the exterior_algebra, which maps tensor_algebra.ι R x to exterior_algebra.ι R x.

Equations
@[simp]
theorem tensor_algebra.to_exterior_ι {R : Type u1} {M : Type u2} [ M] (m : M) :