Proposition 7

Suppose that H is an $H_0$-algebra of finite type. Then the function $p \mapsto \dim(H \otimes_{H_0} \kappa(p))$ is upper semicontinuous on $\mathrm{Spec}(H_0)$.

Since H is a finitely generated $H_0$-algebra, each $H_i$ is a finitely generated $H_0$-module (III, § 1, No. 2, corollary to Prop. 1) and H is generated as an $H_0$-algebra by $H_0 \oplus H_1 \oplus \cdots \oplus H_r$ for a suitable integer $r \geq 0$. Let $p \in \mathrm{Spec}(H_0)$ and put $\dim(H \otimes_{H_0} \kappa(p)) = n \geq 0$. By Corollary 1 to Prop. 6, there exist elements $a_1, ..., a_n$ of H, all homogeneous of the same degree $d > 0$, such that the $\kappa(p)$-homomorphism $\overline{\varphi} : \kappa(p)[X_1, ..., X_n] \to H \otimes_{H_0} \kappa(p)$ which maps $X_i$ to $a_i \otimes 1$ for $1 \leq i \leq n$, makes $H \otimes_{H_0} \kappa(p)$ into a finite $\kappa(p)[X_1, ..., X_n]$-algebra. Let $\varphi$ denote the $H_0$-homomorphism of $H_0[X_1, ..., X_n] = R$ into H which maps $X_i$ to $a_i$ for $1 \leq i \leq n$. If, for every $m \in \mathbf{Z}$, we put $H'_m = \sum_{(m-1)d < i \leq md} H_i$, we obtain a $\mathbf{Z}$-grading on H, compatible with the structure of R-module given by $\varphi$. Each $H'_m$ is finitely generated over $H_0$. By Lemma 5, $H_p$ is a finitely generated $R_p$-module. By Lemma 6, there therefore exists $f \in H_0 - p$ such that $H_f$ is a finitely generated $R_f$-module. For every $q \in \mathrm{Spec}(H_0)_f$, $H \otimes_{H_0} \kappa(q)$ is a finite $\kappa(q)[X_1, ..., X_n]$-algebra, therefore $\dim(H \otimes_{H_0} \kappa(q)) \leq n$ (§ 2, No. 3, Theorem 1), which completes the proof.