Corollary 1

Let $A$ be a Zariski ring (III, $§ 3$, No. 3, Definition 2). For every $A$-module $M$ of finite type, one has $\dim_{\hat{A}}(\hat{M}) = \dim_A(M)$.

In fact, the topology of $A$ is the $a$-adic topology, where $a$ is an ideal contained in the radical of $A$ (loc. cit.), that is, contained in every maximal ideal $m$ of $A$. It is therefore enough to apply assertion b) of Proposition 8.