Exercise 9, § 5

Let A be a commutative ring, F the free associative algebra over A of the set of integers $I = \{1, n\}$, where $n \geq 2$, and $X_i$ ($1 \leq i \leq n$) the corresponding indeterminates, so that a canonical basis of F over A consists of the products $X_{i_1} X_{i_2} \ldots X_{i_r}$, where $(i_1, \ldots, i_r)$ is an arbitrary finite sequence of elements of $I$.

(a) Let $C_2$ denote the set of commutators $[X_i, X_j] = X_i X_j - X_j X_i$ for $1 \leq i < j \leq n$. Assuming that $C_{m-1}$ is defined, $C_m$ denotes the set of elements $[X_i, P] = X_i P - P X_i$, where $1 \leq i \leq n$ and P runs through $C_{m-1}$. Let U be the graded subalgebra of F generated by 1 and the union of the $C_m$ for $m \geq 2$. Show that, for $1 \leq i \leq n$, $[X_i, P] \in U$ for all $P \in U$ (confine it to the case where P is a product of elements of $\bigcup_m C_m$ and argue by induction on the number of factors).

(b) Suppose from now on that A is a field. For every integer $m \geq 0$, let $U_m$ be the vector A-space of homogeneous elements of degree m in U (so that $U_0 = A, U_1 = \{0\}$); choose a basis $R_{m,1}, \ldots, R_{m,d_m}$ in each $U_m$, consisting of products of elements of $\bigcup_{k \leq m} C_k$. For every ordered pair of integers $(m, k)$ such that $m \geq 2, 1 \leq k \leq d_m$, let $E_{m, k}$ be the vector sub-A-space of U generated by the $R_{p, j}$ such that $p > m$ or $p = m$ and $j \geq k$; show that $E_{m, k}$ is a two-sided ideal of U and that, if $L_{m, k}$ is the left ideal of F generated by $E_{m, k}$, $L_{m, k}$ is a two-sided ideal of F.

(c) For every multiindex $\alpha = (\alpha_1, \ldots, \alpha_n) \in \mathbf{N}^n$, we write

Show that the elements $X^\alpha R_{m, k}$ ($m > 0, 1 \leq k \leq d_m$) form a basis of F over K. (To prove that these elements form a generating system of the vector space F, prove that every element $X_{i_1} \ldots X_{i_r}$ is a linear combination of them by induction on the degree r. To see that the $X^\alpha R_{m, k}$ are linearly independent, argue by reductio ad absurdum by considering the maximum degree with respect to one of the $X_i$ of the monomials $X^\alpha$ which would appear in a linear relation between the $X^\alpha R_{m, k}$ and proceed by induction on this maximum degree; for this, reduce it to the case where K is infinite (considering if necessary $F \otimes_K K'$ for an extension field $K'$ of K) and observe that when $X_i + \lambda$ is substituted in an $R_{m, k}$ for $X_i$, for some $\lambda \in K'$, the element $R_{m, k}$ is unaltered).

(d) Show that the two-sided ideal $\mathfrak{J}(F)$ of F generated by the commutators $[u, v] = uv - vu$ of elements of F is the sum of the $U_m$ for $m \geq 2$; $F/\mathfrak{J}(F)$ is isomorphic to $A[X_1, \ldots, X_n]$ and hence is an integral domain.

(e) Show that the algebra U is not a free associative algebra by proving that the quotient $U/\mathfrak{J}(U)$, where $\mathfrak{J}(U)$ is generated by the commutators of elements of U, admits nilpotent elements $\neq 0$. (Consider the image $z'$ in $U/\mathfrak{J}(U)$ of the element $z = [X_i, X_j]$ of U; ${z'}^2 \neq 0$ but ${z'}^4 = 0$.)