Graded modules/algebras on commutative monoids?

Question

In the book Algebra of the Bourbaki group they deal with graded modules/algebras which are graded on a commutative monoid.

What is the need to require the commutativity condition?

Thanks.

Answer 1

Let $M$ be any monoid. This determines a monoidal category of $M$-graded abelian groups $A_m, m \in M$, with monoidal structure given by the convolution product

$$(A \otimes B)_m = \sum_{m_1 m_2 = m} A_{m_1} \otimes B_{m_2}$$

and an $M$-graded ring is a monoid in this monoidal category. (This is a special case of Day convolution.)

If $M$ is commutative, then the category of $M$-graded abelian groups is symmetric monoidal, and now you can define a commutative $M$-graded ring to be a commutative monoid in this symmetric monoidal category. Otherwise there's no way to state commutativity entirely in the category of $M$-graded abelian groups. The basic problem is that if $x \in A_m$ and $x' \in A_{m'}$ then $x x' \in A_{m m'}$ but $x' x \in A_{m' m}$ so they can't be compared if $m m' \neq m' m$.