Proposition 2

Let A be a local ring and B an inflation of A.

a) The ring B is local and one has $m_A B = m_B$.

b) The A-algebra B is faithfully flat.

c) The canonical homomorphism

is bijective.

d) If A is noetherian, the same is true of B and the Hilbert-Samuel series (VIII, § 4, No. 3) of A and B are equal.

Let $(B_\lambda)_{\lambda \in \Lambda}$ be a family of subalgebras of B satisfying conditions a), b) and c) of Def. 2.

Let $\Lambda'$ be the set of indices $\lambda \in \Lambda$ such that, for every $\mu \leq \lambda$ in $\Lambda$, the A-algebra $B_\mu$ is local and faithfully flat, and such that one has $m_{B_\mu} = m_A B_\mu$. Suppose that $\Lambda' \neq \Lambda$ and let $v$ be the smallest element of $\Lambda - \Lambda'$. One has $\alpha \in \Lambda'$, hence $v \neq \alpha$. Now $S_v$ is contained in $\Lambda'$. If $S_v$ has no greatest element, one has $B_v = \bigcup_{\lambda \in S_v} B_\lambda$ and $v$ belongs to $\Lambda'$ by Prop. 1 of No. 1. If $S_v$ has a greatest element $\mu$, one has $\mu \in \Lambda'$ and $B_v$ is an elementary inflation of $B_\mu$: one again has $v \in \Lambda'$ by the remarks following Def. 1, whence a contradiction.

When A is noetherian, one proves analogously that the set $\Lambda''$ of indices $\lambda \in \Lambda$ such that the ring $B_\lambda$ is noetherian is equal to $\Lambda$.

Thus one has $\omega \in \Lambda'$, whence assertions a) and b). When A is noetherian, one has $\omega \in \Lambda''$, hence $B = B_\omega$ is noetherian.

Assertion c) follows from a), b), and Theorem 1 of III, § 5, No. 2. Suppose that A (hence B) is noetherian; since one has

for every $n \in \mathbf{N}$, the Hilbert-Samuel series of A and B are equal.