On Associativity of Graded Algebras

Necessary and sufficient conditions are given to guarantee associativity of algebras and co-algebras. As main application, it is discussed the associativity of the tensor algebra T (V ) of a free module V over any commutative ring R. 1. Review of basic definitions Recall that a vector space A, or more generally a (free) module over a commutative ring R is an R-algebra if a multiplication map mA : A⊗A → A and a unit map ηA : R → A are defined with the following properties: mA,ηA are R-linear; (1) mA(idA⊗mA) = mA(mA ⊗ idA) (associativity); (2) mA(idA⊗ηA) = mA(ηA ⊗ idA) ∼= idA (unit). (3) (2) A⊗A⊗A mA⊗id −→ A⊗A   yidA⊗mA   ymA A⊗A mA −→ A (3) R⊗A ηA⊗idA −→ A⊗A idA⊗ηA ←− A⊗R λց   ymA ւρ A where we denote by λ : R⊗A ∼ −→ A and by ρ : A⊗R ∼ −→ A the natural isomorphisms called left unit constraintand respectively right unit constraint. Throughout this paper ⊗ means ⊗R. Note that (2) expresses the associativity of m: (a ·b) · c = a · (b · c) ⇐⇒ mA(mA(a⊗b)⊗ c) = mA(a⊗mA(b⊗ c)) ⇐⇒ mA ◦ (mA ⊗ idA)((a⊗b)⊗ c) = = mA ◦ (idA⊗mA)(a⊗ (b⊗ c)) ∀a,b,c ∈ A ⇐⇒ mA(mA ⊗ idA) = mA(idA⊗mA). (3) says that A has a unit element 1A = ηA(1) =identity element: a ·1A = a ·ηA(1) = mA(idA⊗ηA)(a⊗1) = 1A ⊗a = = mA(ηA ⊗ idA)(1⊗a) λ = a ρ = a ∀a ∈ A. ∗Partially supported by MURST.