Exercise 13, § 2

Let A be a commutative ring and E an A-module. An A-module $E_n$ is defined inductively by writing

We write $ME = \bigoplus_{n=1}^\infty E_n$ and an A-algebra structure is defined on ME by setting, for $x_p \in E_p$ and $x_q \in E_q$, $x_p x_q = x_p \otimes x_q \in E_{p+q}$.

(a) Show that for every A-algebra B and every A-linear mapping $f : E \to B$, there exists one and only one A-homomorphism of algebras $ME \to B$ extending $f$. ME is called the free algebra of the A-module E.

(b) An integer $a(n)$ is defined by induction on $n$ by the formulae

Show that $E_n$ is isomorphic to the direct sum of $a(n)$ modules isomorphic to $E^{\otimes n}$.

(c) Let $f(T)$ be the formal power series $\sum_{n=1}^\infty a(n)T^n$. Show that

Deduce the formulae

(d) If I is any set and $E = A^{(I)}$, show that $Lib_A(I)$ is identified with ME. There are canonical homomorphisms $M(I) \xrightarrow{\omega} N^{(I)} \xrightarrow{l} N$ whose composition is length in $M(I)$; $\omega(m)$ is called the multidegree of $m \in M(I)$; the mapping $\omega$ defines on $Lib_A(I)$ a graduation of type $N^{(I)}$. Show that the set $(Lib_A(I))_\alpha$ of homogeneous elements of multidegree $\alpha \in N^{(I)}$ under this graduation is a free A-module of rank equal to

where $\alpha! = \prod_{i \in I} (\alpha(i))!$ and $n = \sum_{i \in I} \alpha(i) = l(\alpha)$ is the length of $\alpha$.