mathlib documentation

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 #

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) [comm_semiring R] (M : Type u2) [add_comm_monoid M] [module R M] :
tensor_algebra R Mtensor_algebra R M → Prop

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) [comm_semiring R] (M : Type u2) [add_comm_monoid M] [module R M] :
Type (max u1 u2)

The exterior algebra of an R-module M.

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

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

Equations
@[simp]
theorem exterior_algebra.ι_sq_zero {R : Type u1} [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R M] (m : M) :

As well as being linear, ι m squares to zero

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

Equations
@[simp]
theorem exterior_algebra.lift_symm_apply (R : Type u1) [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R M] {A : Type u_1} [semiring A] [algebra R A] (F : exterior_algebra R M →ₐ[R] A) :
@[simp]
theorem exterior_algebra.ι_comp_lift (R : Type u1) [comm_semiring R] {M : Type u2} [add_comm_monoid 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) :
@[simp]
theorem exterior_algebra.lift_ι_apply (R : Type u1) [comm_semiring R] {M : Type u2} [add_comm_monoid 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) :
@[simp]
theorem exterior_algebra.lift_unique (R : Type u1) [comm_semiring R] {M : Type u2} [add_comm_monoid 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 : exterior_algebra R M →ₐ[R] A) :
@[simp]
theorem exterior_algebra.lift_comp_ι {R : Type u1} [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R M] {A : Type u_1} [semiring A] [algebra R A] (g : exterior_algebra R M →ₐ[R] A) :
@[ext]
theorem exterior_algebra.hom_ext {R : Type u1} [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R M] {A : Type u_1} [semiring A] [algebra R A] {f g : exterior_algebra R M →ₐ[R] A} (h : f.to_linear_map.comp (exterior_algebra.ι R) = g.to_linear_map.comp (exterior_algebra.ι R)) :
f = g

See note [partially-applied ext lemmas].

theorem exterior_algebra.induction {R : Type u1} [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R M] {C : exterior_algebra R M → Prop} (h_grade0 : ∀ (r : R), C ((algebra_map R (exterior_algebra R M)) r)) (h_grade1 : ∀ (x : M), C ((exterior_algebra.ι R) x)) (h_mul : ∀ (a b : exterior_algebra R M), C aC bC (a * b)) (h_add : ∀ (a b : exterior_algebra R M), C aC bC (a + b)) (a : exterior_algebra R 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.ι_inv {R : Type u1} [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R 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
@[simp]
theorem exterior_algebra.ι_add_mul_swap {R : Type u1} [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R M] (x y : M) :
theorem exterior_algebra.ι_mul_prod_list {R : Type u1} [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R M] {n : } (f : fin n → M) (i : fin n) :
((exterior_algebra.ι R) (f i)) * (list.of_fn (λ (i : fin n), (exterior_algebra.ι R) (f i))).prod = 0
def exterior_algebra.ι_multi (R : Type u1) [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R M] (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} [comm_semiring R] {M : Type u2} [add_comm_monoid M] [module R M] {n : } (v : fin n → M) :