Theorem 12.31.43B7C9C492833

Proof. Write $R_{r,k}$ for the component of tensor degree $r$ and weight $-k$. By the preceding results only $k\geq 0$ occurs, multiplication adds both indices, and the subalgebra of weight zero is $\mathbb C[h]$. Let $I$ be the ideal formed by all components with $k>0$, and put

Its component of weight zero is $\mathbb C$, and its ideal $J$ of positive weight index satisfies $J^2=0$, so $S=\mathbb C\oplus J$ as a vector space.

For every $r\geq 1$, products in $I^2$ have weight index at least two. Consequently the component of $S$ of degree $r$ and weight index one is

For $r=1$ this holds because $R_{0,1}=0$, and for $r\geq 2$ it is the last assertion of lemma 12.31.2 with $s=r-1$. Each of these components has dimension $d>0$, and they lie in distinct degrees, so $J$ is infinite dimensional.

If $R$ were finitely generated, its quotient $S$ would be finitely generated too. Write a finite set of algebra generators of $S$ as $c_ i+j_ i$, $1\leq i\leq N$, with $c_ i\in \mathbb C$ and $j_ i\in J$. Because $J^2=0$, every polynomial in these generators lies in the finite-dimensional span of $1,j_1,\ldots ,j_ N$. This contradicts the infinite dimension of $J$. $\square$