Corollary 2

If $\mathbf{A} = \bigoplus_{i \in \mathbf{Z}} \mathbf{A}_i$ is an integrally closed graded domain, the domain $\mathbf{A}_0$ is integrally closed.

The field of fractions $K_0$ of $\mathbf{A}$, is identified (in the notation of Proposition 21) with a subring of the ring of homogeneous elements of degree 0 of $S^{-1}A$; every element of $K_0$ integral over $A$, (and a fortiori over $A$) belongs therefore by hypothesis to $A$.