Proposition 21

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 $\mathbf{A}$. The integral closure $\mathbf{A}'$ of $\mathbf{A}$ is then a graded subring of $S^{-1}\mathbf{A}$. If further $\mathbf{A}_i = 0$ for $i < 0$, then $\mathbf{A}'_i = 0$ for $i < 0$.

If $\mathbf{A} = \mathbf{A}_0$, the proposition is trivial. Otherwise we may apply Lemma 4; the ring $S^{-1}\mathbf{A}$ is an integrally closed domain and therefore $\mathbf{A} \subset S^{-1}\mathbf{A}$; as $S^{-1}\mathbf{A}$ is graded, so is $\mathbf{A}'$ by Proposition 20; the latter assertion also follows from Proposition 20.