Remark

(1) Let I be an infinite indexing set and $\Delta_0$ a commutative monoid; let $(\Delta_i)_{i \in I}$ denote the family such that $\Delta_i = \Delta_0$ for all $i$ and suppose given for every ordered pair of distinct indices $(i, j)$ of I a mapping $\varepsilon_{ij} : \Delta_i \times \Delta_j \to A$ satisfying conditions (22), (23) and (24) (no. 7); this will also be called a system of commutation factors over the family $(\Delta_i)$. Consider a family $(E_i)_{i \in I}$ of graded A-algebras of type $\Delta_0$; for each finite subset J of I, let $E_J$ denote a graded tensor $\varepsilon$-product of type $\Delta_0$ of the subfamily $(E_i)_{i \in J}$ (with an arbitrary choice of a total ordering on J). If J, J' are two finite subsets of I such that $J \subset J'$, a canonical homomorphism of graded algebras of type $\Delta_0$, h_{J'J}: E_J \to E_{J'}, has been defined above and the uniqueness properties of these homomorphisms show immediately that if J \subset J' \subset J'' are three finite subsets of I, then h_{J''J} = h_{J'J'} \circ h_{J'J}. Thus there is a direct system (E_J, h_{J'J}) of graded algebras of type $\Delta_0$ (§ 3, no. 3), whose indexing set is the right directed set $\mathfrak{F}(I)$ of finite subsets of I. The graded algebra of type $\Delta_0$, the direct limit of this direct system (§ 3, no. 3), is called a graded tensor $\varepsilon$-product of type $\Delta_0$ of the family $(E_i)_{i \in I}$; it is also denoted by $\varepsilon \bigotimes_{i \in I} E_i$. When all the $\Delta_i$ are equal to $\mathbf{Z}$ and $\varepsilon_{ij}(a_i, \beta_j) = (-1)^{\alpha_i \beta_j}$, the tensor product $\varepsilon \bigotimes_{i \in I} E_i$ is also called the skew tensor product of the family $(E_i)_{i \in I}$ and is denoted by $g \bigotimes_{i \in I} E_i$. We leave to the reader the task of formulating and proving the proposition which generalizes Proposition 10 of no. 7 to the case where I is infinite, as Proposition 8 of no. 5 generalizes Proposition 5 of no. 2 to the case where I is infinite. Note that the underlying A-module of $\varepsilon \bigotimes_{i \in I} E_i$ is the same as that underlying the (non-graded) tensor product of the family $(E_i)_{i \in I}$ of non-graded algebras defined in no. 5.

(2) Let E be a graded A-algebra of type $\Delta_0$ (where $\Delta_0$ is a commutative monoid) and $\rho : A \to B$ a ring homomorphism; the graduation on $\rho^*(E)$ (II, § 11, no. 5) is identical with the graduation on the graded tensor product $B \otimes_A E$, where B has the trivial graduation.