Proposition 8

Let $A$ be a noetherian ring, $a$ an ideal of $A$, $M$ a finitely generated $A$-module, $\hat{A}$ and $\hat{M}$ the separated completions of $A$ and $M$ respectively for the $a$-adic topology.

a) Let $m$ be a prime ideal of $A$ containing $a$ and $\hat{m} = m\hat{A}$. Then $\hat{m}$ is a prime ideal of $\hat{A}$ and one has $\dim_{\hat{A}_{\hat{m}}}(\hat{M}_{\hat{m}}) = \dim_{A_m}(M_m)$.

b) One has $\dim_{\hat{A}}(\hat{M}) = \sup_m \dim_{A_m}(M_m)$, where $m$ ranges over the set of prime (resp. maximal) ideals of $A$ containing $a$. In particular, one has $\dim_{\hat{A}}(\hat{M}) \leqslant \dim_A(M)$.

a) Since $\hat{A}/\hat{m}$ is identified with $A/m$, $\hat{m}$ is a prime ideal of $\hat{A}$. By Theorem 3 of III, $§ 3$, No. 4, $\hat{A}$ is flat over $A$, hence $\hat{A}_{\hat{m}}$ is flat over $A_m$. Moreover the canonical mapping of $A$ into $\hat{A}$ induces an isomorphism of $A/a$ onto $\hat{A}/a\hat{A}$, hence also an isomorphism of $A_m/mA_m$ onto $\hat{A}_{\hat{m}}/m\hat{A}_{\hat{m}}$. The conclusion follows by applying Proposition 7 to the rings $A_m$ and $\hat{A}_{\hat{m}}$ and to the modules $M_m$ and $\hat{A}_{\hat{m}}$ since $M_m \otimes_{A_m} \hat{A}_{\hat{m}}$ is isomorphic to $\hat{M}_{\hat{m}}$ (III, loc. cit. and Proposition 8).

b) By Proposition 8 of III, $§ 3$, No. 4, the mapping $m \mapsto \hat{m}$ is a bijection of the set of maximal ideals of $A$ containing $a$ onto the set of maximal ideals of $\hat{A}$. Assertion b) follows from this and from Proposition 9 of $§ 1$, No. 4.