Proposition 10

Let $\mathbf{A}$ be a commutative ring and $(\Delta_i)_{i \in I}$ a finite family of commutative monoids written additively; for each $i \in I$ let $E_i$ be a graded $\mathbf{A}$-algebra of type $\Delta_i$. Finally, let $(\varepsilon_{ij})$ be a system of commutation factors over the $\Delta_i$ with values in $\mathbf{A}$. Then there exist a graded $\mathbf{A}$-algebra $E$ of type $\Delta = \prod_{i \in I} \Delta_i$ and for each $i \in I$ an algebra homomorphism $h_i : E_i \to E$, with the following properties:

(i) If $\phi_i : \Delta_i \to \Delta$ is the canonical homomorphism, then $h_i$ is a graded homomorphism (II, § 11, no. 2), in other words, $h_i(E_i^{\alpha_i}) \subset E^{\phi_i(\alpha_i)}$, where $(E_i^{\alpha_i})$ and $(E^{\alpha})$ denote the respective graduations on $E_i$ and $E$.

(ii) If $i \neq j$ and $x_i$ (resp. $x_j$) is a homogeneous element of $E_i$ (resp. $E_j$) of degree $\alpha_i \in \Delta_i$ (resp. $\beta_j \in \Delta_j$), then

(iii) For every $\mathbf{A}$-algebra $F$ and every system of homomorphisms $f_i : E_i \to F$ satisfying the conditions

where $i, j, x_i, x_j, \alpha_i, \beta_j$ are as in (ii), then there exists one and only one algebra homomorphism $f : E \to F$ such that $f_i = f \circ h_i$ for all $i \in I$. Moreover the underlying $\mathbf{A}$-module of $E$ is the tensor product $\bigotimes_{i \in I} E_i$.

Consider the $\mathbf{A}\text{-module}$ $E = \bigotimes_{i \in I} E_i$; it is identified with the direct sum of the submodules $E^{\alpha}$, where, for each $\alpha = (\alpha_i) \in \Delta$, we write $E^{\alpha} = \bigotimes_{i \in I} E_i^{\alpha_i}$; the $E^{\alpha}$ therefore form a graduation of type $\Delta$ on the $\mathbf{A}$-module $E$. We shall define on $E$ a graded $\mathbf{A}$-algebra structure of type $\Delta$. For this let $I$ be given a total ordering; for every ordered pair of elements $\alpha = (\alpha_i), \beta = (\beta_i)$ of $\Delta$, we must first define an $\mathbf{A}$-bilinear mapping of $E^{\alpha} \times E^{\beta}$ into $E^{\alpha + \beta}$, or alternatively an $\mathbf{A}$-linear mapping $m_{\alpha \beta}$ of $E^{\alpha} \otimes_{\mathbf{A}} E^{\beta}$ into $E^{\alpha + \beta}$. We shall define $m_{\alpha \beta}$ by the condition

for $x_i \in E_i^{\alpha_i}, y_i \in E_i^{\alpha_i}$, where

The right hand side of (27) obviously belongs to $E^{\alpha + \beta}$ and the mapping $(x_1, \ldots, x_n, y_1, \ldots, y_n) \mapsto \varepsilon(\alpha, \beta) \bigotimes_{i \in I} (x_i y_i)$ is $\mathbf{A}$-multilinear in the product of the $E_i^{\alpha_i}$ and the $E_i^{\beta_i}$ ($1 \leq i \leq n$). Then it must be proved that the multiplication thus defined on E is associative; now, if $\gamma = (\gamma_i)$ is a third element of $\Delta$ and $z_i \in E_i^{\gamma_i'}$ for $1 \leq i \leq n$, then

and it reduces to verifying the identity

But the latter follows immediately from the relations

themselves immediate consequences of the definition (28) and (22) and (23).

If, for all $i \in I$, $e_i$ denotes the unit element of $E_i$, we know that $e_i$ is homogeneous of degree 0 (§ 3, no. 1), hence $e = \bigotimes_{i \in I} e_i$ is homogeneous of degree 0 and it follows from (27), (28) and the relations

that $e$ is unit element of $E$, which completes the definition on $E$ of the desired graded A-algebra structure. Then take $h_i(x_i) = x_i \otimes \bigotimes_{j \neq i} e_j$; to verify that $h_i(x_i x_i') = h_i(x_i) h_i(x_i')$ for $x_i, x_i'$ in $E_i$, attention may be confined to the case where $x_i$ and $x_i'$ are homogeneous and then this relation follows immediately from (27) and the relations $\varepsilon_{ij}(\alpha_i, 0) = \varepsilon_{ij}(0, \beta_j) = 1$; the same relations and (24) prove also that the $h_i$ satisfy conditions (i) and (ii) of the statement and that

where the right hand side is the product of the ordered sequence $(h_i(x_i))_{i \in I}$ in E with the given total ordering on I (I, § 1, no. 2) (it suffices to argue by induction on the number of $x_i$ (assumed homogeneous) distinct from the $e_i$).

It remains to prove condition (iii); note that the mapping

where the right hand side is the product of the ordered sequence $(f_i(x_i))_{i \in I}$ with the given total ordering on $I$, is A-multilinear. Then there exists one and only one A-linear mapping $f : E \to F$ such that

Clearly $f(e)$ is the unit element of $F$ and $f \circ h_i = f_i$; to verify that $f$ is an algebra homomorphism, in other words that $f(x)f(y) = f(xy)$ for $x, y$ in $E$, attention may be confined, by linearity, to the case where $x = \bigotimes_{i \in I} x_i$ and $y = \bigotimes_{i \in I} y_i$, $x_i$ (resp. $y_i$) being homogeneous of degree $\alpha_i$ (resp. $\beta_i$) for all $i \in I$. The relation to be verified then reduces, by (27), to

But, taking account of relations (26), this is a consequence of Lemma 2 of no. 6.

Clearly the algebra $E$ and the canonical mapping $\psi : \bigotimes_{i \in I} E_i \to E$ constitute a solution of the universal mapping problem (Set Theory, IV, § 3, no. 1), where $\Sigma$ is the species of A-algebra structure and the $\alpha$-mappings $\prod_i f_i$ from $\prod_i E_i$ to an A-algebra, satisfying conditions (26).

For a fixed total ordering on $I$, the graded algebra $E$ defined in the proof of Proposition 10 will be called a graded tensor $\varepsilon$-product of type $\Delta$ of the family $(E_i)_{i \in I}$ of graded algebras of type $\Delta_i$ and will be denoted by $^{\varepsilon} \bigotimes_{i \in I} E_i$ (if no confusion can arise over the ordering on $I$); similarly, the homomorphism $f : E \to F$ defined in the proof of Proposition 10 will be denoted by $^{\varepsilon} \bigotimes_{i \in I} f_i$. The homomorphisms $h_i$ are called canonical. We also write $^{\varepsilon} G^{\otimes n}$ when $I = \{1, n\}$ and all the $E_i$ are equal to the same algebra $G$.