Lemma 1

Let A be a semi-local noetherian ring and B a finite A-algebra. Then the ring B is semi-local and noetherian; let m₁, ..., mₙ be its maximal ideals.

The canonical homomorphism of B into $\prod_{i=1}^n \hat{B}_{m_i}$ extends to an isomorphism of $\hat{A} \otimes_A B$ onto $\prod_{i=1}^n \hat{B}_{m_i}$.

By IV, § 2, No. 5, cor. 3 to prop. 9, the ring B is semi-local and $m_A B$ is an ideal of definition of it. By III, § 3, No. 4, th. 3, (ii), the ring $\hat{A} \otimes_A B$ is the completion of B for the topology defined by its radical; one then applies III, § 2, No. 13, corollary to prop. 19.