Lemma 4

Let $\mathbf{A} = \bigoplus_{i \in \mathbf{Z}} \mathbf{A}_i$ be a graded integral domain and $S$ the set of homogeneous elements $\neq 0$ of $\mathbf{A}$.

(i) Every homogeneous element $\neq 0$ of $S^{-1}\mathbf{A}$ is invertible, the ring $K_0 = (S^{-1}\mathbf{A})_0$ is a field and the set of $i \in \mathbf{Z}$ such that $(S^{-1}\mathbf{A})_i \neq 0$ is a subgroup $q\mathbf{Z}$ of $\mathbf{Z}$ (where $q \geqslant 0$).

(ii) Suppose that $q \geqslant 1$ and let $t$ be a non-zero element of $(S^{-1}\mathbf{A})_q$. Then the $K_0$-homomorphism $f$ of the polynomial ring $K_0[X]$ to $S^{-1}\mathbf{A}$ which maps $X$ to $t$ extends to an isomorphism of $K_0[X, X^{-1}]$ onto $S^{-1}\mathbf{A}$ and $S^{-1}\mathbf{A}$ is integrally closed.

The assertions in (i) follow immediately from the definitions and the hypothesis that $\mathbf{A}$ is an integral domain, for if $a/s$ and $a'/s'$ are two homogeneous elements $\neq 0$ of $S^{-1}\mathbf{A}$ of degrees $i$ and $i'$, $aa'/ss'$ is a homogeneous element $\neq 0$ and of degree $i + i'$. To show (ii), we note that since $t$ is invertible in $S^{-1}\mathbf{A}$ the homomorphism $f$ extends in a unique way to a homomorphism $\tilde{f}: K_0[X, X^{-1}] \to S^{-1}\mathbf{A}$ and necessarily $\tilde{f}(X^{-1}) = t^{-1}$. On the other hand, by definition of $q$, every homogeneous element $\neq 0$ of $S^{-1}\mathbf{A}$ is of degree $qn \ (n \in \mathbf{Z})$ and hence can be written uniquely in the form $\lambda t^n$ where $\lambda \in K_0$ (since $S^{-1}\mathbf{A}$ is an integral domain); hence $\tilde{f}$ is bijective. Finally, we know that $K_0[X]$ is integrally closed (no. 3, Proposition 10) and hence so is $K_0[X, X^{-1}]$ (no. 5, Corollary 1 to Proposition 16), which completes the proof of the Lemma.