Theorem 1

Let $f : A \to A'$ be a surjective local homomorphism of local rings and let $B'$ be an inflation of $A'$. There exists an inflation B of A and an isomorphism of A-algebras of $B \otimes_A A'$ onto $B'$. A) Suppose that $B'$ is an elementary inflation of $A'$. Let us distinguish two cases: 1) If $B'$ is finite over $A'$, choose an isomorphism of $A'-algebras$ $\varphi : A'[X]/(P') \to B'$, where $P' \in A'[X]$ is a monic polynomial whose image is irreducible in $\kappa_{A'}[X]$. Choose a monic polynomial $P \in A[X]$ whose image in $A'[X]$ is $P'$. It is necessarily irreducible modulo the maximal ideal of A. Put then $B = A[X]/(P)$. The A-algebra B is an elementary inflation of A and $\varphi$ induces an isomorphism of A-algebras of $B \otimes_A A'$ onto $B'$. 2) If $B'$ is not finite over $A'$, choose an isomorphism of $A'-algebras$ $\psi : A'[X] \to B'$. Put $B = A[X][$. The A-algebra B is an elementary inflation of A, and $B \otimes_A A'$ is canonically isomorphic to $A'[X][$. Hence $\psi$ induces an isomorphism of A-algebras of $B \otimes_A A'$ onto $B'$. B) We pass to the general case. Let $(B'_\lambda)_{\lambda \in \Lambda}$ be a family of sub-$A'$-algebras of $B'$ having with respect to $A'$ and $B'$ properties a), b), c) of Def. 2. We are going to define by transfinite induction an inductive system $(\tilde{B}_\lambda, i_{\mu \lambda})$ indexed by $\Lambda$ of local rings and injective local homomorphisms, and isomorphisms $u_\lambda : \tilde{B}_\lambda \otimes_A A' \to B'_\lambda$ such that, for $\lambda \leq \mu$, $u_\mu \circ (i_{\mu \lambda} \otimes \mathrm{Id}_{A'}) \circ u_\lambda^{-1}$ is the canonical injection of $B'_\lambda$ into $B'_{\mu}$. If $\alpha$ is the smallest element of $\Lambda$, we put $\tilde{B}_\alpha = A, i_{\alpha \alpha} = \mathrm{Id}_A$ and take for $u_\alpha$ the canonical isomorphism $A \otimes_A A' \to A'$. Let $\nu \in \Lambda$, and suppose that $\tilde{B}_\lambda, u_\lambda$ and $i_{\mu \lambda}$ have been constructed when $\lambda \leq \mu < \nu$. Let $S_\nu$ be the set of elements $\varepsilon$ of $\Lambda$ such that $\varepsilon < \nu$. If $S_\nu$ has no greatest element, we take for $\tilde{B}_\nu$ the inductive limit of the $\tilde{B}_\lambda$ for $\lambda \in S_\nu$, for $u_\nu$ the composite isomorphism $\tilde{B}_\nu \otimes_A A' \to \lim (\tilde{B}_\lambda \otimes_A A') \to \lim B'_\lambda \to B'_\nu$, and for $i_{\nu \lambda}$, when $\lambda \in S_\nu$, the canonical mapping of $\tilde{B}_\lambda$ into $\tilde{B}_\nu$. If $S_\nu$ has a greatest element $\mu$, then $B'_\nu$ is an elementary inflation of $B'_{\mu}$. By A), there exists an elementary inflation i_{v\mu}: \tilde{B}_{\mu} \to \tilde{B}_{v} et un isomorphisme de $\tilde{B}_{\mu}$-algèbres de $\tilde{B}_{v} \otimes_{\tilde{B}_{\mu}} B'_{\mu}$ sur $B'_{v}$. We take for $u_{v}$ the composite isomorphism of A-algebras

and for $i_{v\lambda}$, when $\lambda \in S_{v}$, the homomorphism $i_{v\mu} \circ i_{\mu\lambda}$.

Let us then put $B = \tilde{B}_{\omega}$ and, for every $\lambda \in \Lambda$, denote by $B_{\lambda}$ the image of $\tilde{B}_{\lambda}$ under the canonical injection $\tilde{B}_{\lambda} \to B$. The family $(B_{\lambda})_{\lambda \in \Lambda}$ satisfies conditions a), b), c) of Definition 2, and B is an inflation of A. On the other hand, the homomorphism $u_{\omega}$ is an A'-isomorphism of $B \otimes_{A} A'$ onto $B'$.