Proposition 20

*Let $\mathbf{A} = \bigoplus_{i \in \mathbf{Z}} A_i$, be a graded ring and $\mathbf{B}$ a (commutative) graded $\mathbf{A}$-algebra. The set $\mathbf{A}'$ of elements of $\mathbf{B}$ integral over $\mathbf{A}$ is a graded subalgebra of $\mathbf{B}$.

If $A_i = 0$ for $i < 0$ and $\mathbf{B}$ is a reduced ring, then $A'_i = 0$ for $i < 0$.*

The diagram

(where $\rho$ is the homomorphism defining the $\mathbf{A}$-algebra structure on $\mathbf{B}$ and $\rho'$ the homomorphism canonically derived from it) is commutative, as is immediately verified from the definition (5). Let $x$ be an element of $\mathbf{B}$ integral over $\mathbf{A}$; then $j_{\mathbf{B}}(x)$ is integral over $\mathbf{A}[X, X^{-1}]$ (no. 1, Proposition 2) and it therefore follows from no. 5, Proposition 16 that there exists an integer $m > 0$ such that $X^m j_{\mathbf{B}}(x)$ is an element of $\mathbf{B}[X]$ integral over $\mathbf{A}[X]$. We then deduce from no. 3, Proposition 12 that the coefficients of the polynomial $X^m j_{\mathbf{B}}(x)$ are integral over $\mathbf{A}$; as these coefficients are by definition the homogeneous components of $x$, it is seen that these are integral over $\mathbf{A}$, which proves that $\mathbf{A}'$ is a graded subalgebra of $\mathbf{B}$.

Suppose now that $x \in A'_i$ where $i < 0$; the remark at the beginning of this no. shows that $x$ satisfies an equation of the form (4) where $a_k' \in A_{kt}$ for $1 \leq k \leq n$. If $A_j = 0$ for $j < 0$, then $x^n = 0$ and if $\mathbf{B}$ is a reduced ring we conclude that $x = 0$ and hence $A'_i = 0$ for all $i < 0$ in this case.

Recall (Chapter II, § 2, no. 9) that, if $\mathbf{A} = \bigoplus_{i \in \mathbf{Z}} \mathbf{A}_i$ is a graded ring and $S$ is a multiplicative subset of $\mathbf{A}$ consisting of homogeneous elements, a graded ring structure is defined on $S^{-1}\mathbf{A}$ by taking the set $(S^{-1}\mathbf{A})_i$ of homogeneous elements of degree $i$ to be the set of elements of the form $a/s$, where $a \in \mathbf{A}$ and $s \in S$ are homogeneous and such that $\deg(a) - \deg(s) = i$