Proposition 2

Let $A = \bigoplus_{i \in \mathbf{Z}} A_i$ be a graded commutative ring with positive degrees and $M = \bigoplus_{i \in \mathbf{Z}} M_i$ a graded $A$-module. Suppose that $A$ is a finitely generated $A_0$-algebra and $M$ a finitely generated $A$-module. Then, for every ordered pair $(d, k)$ of integers such that $d \geqslant 1, 0 \leqslant k \leqslant d - 1$:

(i) $A^{(d)}$ is a finitely generated $A$-module. (ii) $M^{(d, k)}$ is a finitely generated $A^{(d)}$-module.

Let us show that $A$ is a finitely generated $A^{(d)}$-*module*. Let $(a_i)_{1 \leqslant i \leqslant s}$ be a system of generators of the $A$-algebra $A$ consisting of homogeneous elements. The elements of $A$ (finite in number) of the form $a_1^{\alpha_1} a_2^{\alpha_2} \ldots a_s^{\alpha_s}$ such that $0 \leqslant \alpha_i \leqslant d$ for $1 \leqslant i \leqslant s$ constitute a system of generators of the $A^{(d)}$-module $A$; for every system of integers $n_i \geqslant 0$ $(1 \leqslant i \leqslant s)$, there are positive integers $q_i, r_i$ such that $n_i = q_i d + r_i$ where $r_i < d$ $(1 \leqslant i \leqslant s)$; then

which proves our assertion, for every homogeneous element $x \in A$ satisfies $x^d \in A^{(d)}$. Then, if $M$ is a finitely generated $A$-module, it is also a finitely generated $A^{(d)}$-module; as $M$ is the direct sum of the $M^{(d, k)}$ $(0 < k < d - 1)$, each of the $M^{(d, k)}$ is a finitely generated $A^{(d)}$-module, which proves (ii).

Let us apply the above to the graded $A$-module $m = \bigoplus_{i \geqslant 1} A_i$, which is finitely generated by the Corollary to Proposition 1 of no. 2; it is seen that $m^{(d)}$ is a finitely generated $A^{(d)}$-module; therefore (no. 2, Corollary to Proposition 1) $A^{(d)}$ is a finitely generated $A$-algebra.