Proposition 9

Let K be a commutative ring, A a bigebra over K with counit $\gamma$, and $\mathscr{C}$ an additive set of classes of A-modules with the following properties:

(i) Every A-module of type $\mathscr{C}$ is a projective $(*$or, more generally, flat$*)$ K-module.

(ii) If the A-modules E and F are of type $\mathscr{C}$, then so is the A-module $E\otimes F$.

(iii) The A-module $K_{\gamma}$ defined above is of type $\mathscr{C}$. Then there exists a unique ring structure on the additive group $K(\mathscr{C})$ whose multiplication satisfies

for all A-modules E and F of type $\mathscr{C}$. The unit element of $K(\mathscr{C})$ is $[K_{\gamma}]_{\mathscr{C}}$. If the bigebra A is cocommutative, then the ring $K(\mathscr{C})$ is commutative.

Endowed with the law of composition given by $(E,F)\mapsto$ cl(E $\otimes F$), the set $\mathscr{C}$ is a monoid with unit element cl(K$_{\gamma}$). Consequently, $\mathbf{Z}^{(\mathscr{C})}$ is canonically endowed with the structure of a ring with multiplication characterized by the formula

(13) cl(E) cl(F) = cl(E $\otimes F$)

and unit element cl(K$_{\gamma}$) (III, §2, No. 6, p. 446).

Given an A-module F of type $\mathscr{C}$ and an exact sequence

of A-modules of type $\mathscr{C}$, the sequence

deduced from $(\mathscr{E})$ is exact because F is projective $(*$or, more generally, flat$*)$ over K (II, §3, No. 6, p. 251, Proposition 5 and II, §3, No. 7, p. 257, Corollary 6). In the notation of No. 2, we then have

(14) $r_{\mathscr{E}\otimes F}=r_{\mathscr{E}}$ cl(F).

It follows that the subgroup R of $\mathbf{Z}^{(\mathscr{C})}$ generated by the elements of the form $r_{\mathscr{E}}$ is a right ideal of the ring $\mathbf{Z}^{(\mathscr{C})}$, and one proves likewise that it is a left ideal of $\mathbf{Z}^{(\mathscr{C})}$. By definition, $K(\mathscr{C})$ is the quotient group $\mathbf{Z}^{(\mathscr{C})}/R$; hence there exists a unique ring structure on $K(\mathscr{C})$ whose multiplication satisfies $[E]_{\mathscr{C}}[F]_{\mathscr{C}}= [E\otimes F]_{\mathscr{C}}$ for all A-modules E and F of type $\mathscr{C}$. Its unit element is $[K_{\gamma}]$.

When the bigebra A is cocommutative, the monoid $\mathscr{C}$ is commutative; it follows that the ring $\mathbf{Z}^{(\mathscr{C})}$ and the quotient ring $K(\mathscr{C})$ are commutative.