Example 3

More generally, let $A$ be a commutative ring, $b$ an ideal of $A$ and $A$ be given the $b$-adic filtration. If we write $B=\operatorname{gr}_0(A)$, $F=\operatorname{gr}_1(A)=b/b^2$, we know (Algebra, Chapter III), that the identity mapping of the $B$-module $F$ onto itself can be extended uniquely to a homomorphism $u$ from the symmetric algebra $S(F)$ of $F$ to the $B$-algebra $\operatorname{gr}(A)$; it follows from the definition of $\operatorname{gr}(A)$ that $u$ is a surjective homomorphism of graded algebras; for $n\geq1$, every element of $\operatorname{gr}_n(A)$ is a sum of classes mod $b^{n+1}$ of elements of the form $y=x_1x_2\cdots x_n$, where $x_i\in b$ $(1\leq i\leq n)$; if $\xi_i$ is the class of $x_i$ mod. $b^2$, clearly the class of $y$ mod. $b^{n+1}$ is the element $u(\xi_1)\cdots u(\xi_n)$, whence our assertion. In particular, every system of generators of the $B$-module $F$ is a system of generators of the $B$-algebra $\operatorname{gr}(A)$.

If now $E$ is an $A$-module and $E$ is given the $b$-adic filtration, it is seen similarly that the graded $\operatorname{gr}(A)$-module $\operatorname{gr}(E)$ is generated by $\operatorname{gr}_0(E)=E/bE$. To be precise, the restriction $\phi$ to $\operatorname{gr}(A)\times\operatorname{gr}_0(E)$ of the external law on the $\operatorname{gr}(A)$-module $\operatorname{gr}(E)$ is a $\mathbf{Z}$-bilinear mapping of $\operatorname{gr}(A)\times\operatorname{gr}_0(E)$ to $\operatorname{gr}(E)$; moreover $\operatorname{gr}(A)$ is a $(\operatorname{gr}_0(A),\operatorname{gr}_0(A))$-bimodule and $\operatorname{gr}_0(E)$ a $\operatorname{gr}_0(A)$-module; it is immediately verified that, for $a\in\operatorname{gr}(A)$, $\alpha_0\in\operatorname{gr}_0(A)$, $\xi\in\operatorname{gr}_0(E)$,

and hence $\phi$ defines a surjective $\operatorname{gr}_1(A)$-linear mapping

(11)

which is called canonical.