Proposition 8

Let $m = \bigoplus_{i \geq 1} A_i$; let $(p^{(k)})_{1 \leq k \leq s}$ be a finite family of graded prime ideals of $A$ and a graded ideal $\mathfrak{a} \subset A$ such that $\mathfrak{a} \cap m \not\subset p^{(k)}$ for all $k$; then there exists a homogeneous element $z \in \mathfrak{a} \cap m$ not belonging to any of the $p^{(k)}$.

We argue by induction on $n$, the proposition being trivial for $n = 1$. If there exists an index $j$ such that $\mathfrak{a} \cap m \cap p^{(j)}$ is contained in one of the $p^{(k)}$ of index $k \neq j$, it follows from the induction hypothesis that there is a homogeneous element $z' \in \mathfrak{a} \cap m$ not belonging to any of the $p^{(k)}$ for $k \neq j$ and therefore not belonging to $p^{(j)}$ either and this element solves the problem. Suppose then that for every index $j$, $\mathfrak{a} \cap m \cap p^{(j)}$ is not contained in any of the $p^{(k)}$ of index $k \neq j$; the induction hypothesis therefore implies the existence of a homogeneous element $y_j \in \mathfrak{a} \cap m \cap S_j$ belonging to none of the $S_j$ for $k \# j$; as the $y_j$ are all of degrees $\geq 1$, we may assume by replacing them by suitable powers (since the $p^{(k)}$ are prime) that $y_1$ and $\prod_{j=2}^n y_j$ are of the same degree. Then $z = y_1 + \prod_{j=2}^n y_j$ is homogeneous of degree $\geq 1$ and the same argument as in Chapter II, § 1, no. 1, Proposition 2 shows that $z$ solves the problem.