The Stacks project

Proof. Given a deformation of $P$ we can take the completion of it to get a deformation of the completion; this is clear and we encourage the reader to skip the proof. More precisely, let $(A, Q) \to (k, P)$ be a morphism in $\mathcal{F}$, i.e., an object of $\mathcal{D}\! \mathit{ef}_ P$. Observe that $Q$ is a Noetherian ring: the kernel of the surjective ring map $Q \to P$ is nilpotent and finitely generated and $P$ is Noetherian; apply Algebra, Lemma 10.97.5. Denote $J_ Q \subset Q$ the inverse image of $J$ in $Q$. Let $Q^\wedge$ be the $J_ Q$-adic completion of $Q$. Recall that $Q \to Q^\wedge$ is flat (Algebra, Lemma 10.97.2) and hence $Q^\wedge$ is flat over $A$. The induced map $Q^\wedge \to P^\wedge$ induces an isomorphism $Q^\wedge \otimes _ A k = Q^\wedge \otimes _ Q P = P^\wedge$ by Algebra, Lemma 10.97.1 for example. Hence $(A, Q^\wedge ) \to (k, P^\wedge )$ is the desired object of $\mathcal{D}\! \mathit{ef}_{P^\wedge }$. $\square$