mathlib documentation

linear_algebra.tensor_algebra

Tensor Algebras #

Given a commutative semiring R, and an R-module M, we construct the tensor algebra of M. This is the free R-algebra generated (R-linearly) by the module M.

Notation #

  1. tensor_algebra R M is the tensor algebra itself. It is endowed with an R-algebra structure.
  2. tensor_algebra.ι R is the canonical R-linear map M → tensor_algebra R M.
  3. Given a linear map f : M → A to an R-algebra A, lift R f is the lift of f to an R-algebra morphism tensor_algebra R M → A.

Theorems #

  1. ι_comp_lift states that the composition (lift R f) ∘ (ι R) is identical to f.
  2. lift_unique states that whenever an R-algebra morphism g : tensor_algebra R M → A is given whose composition with ι R is f, then one has g = lift R f.
  3. hom_ext is a variant of lift_unique in the form of an extensionality theorem.
  4. lift_comp_ι is a combination of ι_comp_lift and lift_unique. It states that the lift of the composition of an algebra morphism with ι is the algebra morphism itself.

Implementation details #

As noted above, the tensor algebra of M is constructed as the free R-algebra generated by M, modulo the additional relations making the inclusion of M into an R-linear map.

inductive tensor_algebra.rel (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] :
free_algebra R Mfree_algebra R M → Prop

An inductively defined relation on pre R M used to force the initial algebra structure on the associated quotient.

@[instance]
def tensor_algebra.inhabited (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] :
def tensor_algebra (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] :
Type (max u_1 u_2)

The tensor algebra of the module M over the commutative semiring R.

Equations
@[instance]
def tensor_algebra.semiring (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] :
@[instance]
def tensor_algebra.algebra (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [module R M] :
@[instance]
def tensor_algebra.ring (M : Type u_2) [add_comm_monoid M] {S : Type u_1} [comm_ring S] [module S M] :
Equations
def tensor_algebra.ι (R : Type u_1) [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] :

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

Equations
@[simp]
theorem tensor_algebra.lift_symm_apply (R : Type u_1) [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] (F : tensor_algebra R M →ₐ[R] A) :
def tensor_algebra.lift (R : Type u_1) [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] :

Given a linear map f : M → A where A is an R-algebra, lift R f is the unique lift of f to a morphism of R-algebras tensor_algebra R M → A.

Equations
@[simp]
theorem tensor_algebra.ι_comp_lift {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] (f : M →ₗ[R] A) :
@[simp]
theorem tensor_algebra.lift_ι_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] (f : M →ₗ[R] A) (x : M) :
@[simp]
theorem tensor_algebra.lift_unique {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] (f : M →ₗ[R] A) (g : tensor_algebra R M →ₐ[R] A) :
@[simp]
theorem tensor_algebra.lift_comp_ι {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] (g : tensor_algebra R M →ₐ[R] A) :
@[ext]
theorem tensor_algebra.hom_ext {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {A : Type u_3} [semiring A] [algebra R A] {f g : tensor_algebra R M →ₐ[R] A} (w : f.to_linear_map.comp (tensor_algebra.ι R) = g.to_linear_map.comp (tensor_algebra.ι R)) :
f = g

See note [partially-applied ext lemmas].

theorem tensor_algebra.induction {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] {C : tensor_algebra R M → Prop} (h_grade0 : ∀ (r : R), C ((algebra_map R (tensor_algebra R M)) r)) (h_grade1 : ∀ (x : M), C ((tensor_algebra.ι R) x)) (h_mul : ∀ (a b : tensor_algebra R M), C aC bC (a * b)) (h_add : ∀ (a b : tensor_algebra R M), C aC bC (a + b)) (a : tensor_algebra R M) :
C a

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

def tensor_algebra.algebra_map_inv {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] :

The left-inverse of algebra_map.

Equations
def tensor_algebra.ι_inv {R : Type u_1} [comm_semiring R] {M : Type u_2} [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
def free_algebra.to_tensor {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] :

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

Equations
@[simp]
theorem free_algebra.to_tensor_ι {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M] [module R M] (m : M) :