# mathlibdocumentation

linear_algebra.exterior_algebra

# Exterior Algebras

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

## Notation

The exterior algebra of the R-semimodule 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 semimodule 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.

## 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 u_1) (M : Type u_2) [ M] :
→ Prop
• of : ∀ (R : Type u_1) [_inst_1 : (M : Type u_2) [_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 u_1) (M : Type u_2) [ M] :
Type (max u_1 u_2)

The exterior algebra of an R-semimodule M.

Equations
@[instance]
def exterior_algebra.inst (R : Type u_1) (M : Type u_2) [ M] :
M)

@[instance]
def exterior_algebra.inhabited (R : Type u_1) (M : Type u_2) [ M] :

@[instance]
def exterior_algebra.semiring (R : Type u_1) (M : Type u_2) [ M] :

@[instance]
def exterior_algebra.ring {M : Type u_2} {S : Type u_1} [comm_ring S] [ M] :
ring M)

Equations
def exterior_algebra.ι (R : Type u_1) {M : Type u_2} [ M] :

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

Equations
@[simp]
theorem exterior_algebra.ι_square_zero {R : Type u_1} {M : Type u_2} [ M] (m : M) :
( m) * m = 0

As well as being linear, ι m squares to zero

@[simp]
theorem exterior_algebra.comp_ι_square_zero {R : Type u_1} {M : Type u_2} [ M] {A : Type u_3} [semiring A] [ A] (g : →ₐ[R] A) (m : M) :
(g ( m)) * g ( m) = 0

def exterior_algebra.lift (R : Type u_1) {M : Type u_2} [ M] {A : Type u_3} [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 u_1) {M : Type u_2} [ M] {A : Type u_3} [semiring A] [ A] (F : →ₐ[R] A) :
.symm) F = , _⟩

@[simp]
theorem exterior_algebra.ι_comp_lift (R : Type u_1) {M : Type u_2} [ M] {A : Type u_3} [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 u_1) {M : Type u_2} [ M] {A : Type u_3} [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 u_1) {M : Type u_2} [ M] {A : Type u_3} [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 u_1} {M : Type u_2} [ M] {A : Type u_3} [semiring A] [ A] (g : →ₐ[R] A) :
, _⟩ = g

@[ext]
theorem exterior_algebra.hom_ext {R : Type u_1} {M : Type u_2} [ M] {A : Type u_3} [semiring A] [ A] {f g : →ₐ[R] A} (h : = ) :
f = g