Lemma 93.8.8. In Example 93.8.1 let $P$ be a $k$-algebra. Let $J \subset P$ be an ideal. Denote $(P^ h, J^ h)$ the henselization of the pair $(P, J)$. There is a natural functor
of deformation categories.
Proof. Given a deformation of $P$ we can take the henselization of it to get a deformation of the henselization; 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 $\mathcal{D}\! \mathit{ef}_ P$. Denote $J_ Q \subset Q$ the inverse image of $J$ in $Q$. Let $(Q^ h, J_ Q^ h)$ be the henselization of the pair $(Q, J_ Q)$. Recall that $Q \to Q^ h$ is flat (More on Algebra, Lemma 15.12.2) and hence $Q^ h$ is flat over $A$. By More on Algebra, Lemma 15.12.7 we see that the map $Q^ h \to P^ h$ induces an isomorphism $Q^ h \otimes _ A k = Q^ h \otimes _ Q P = P^ h$. Hence $(A, Q^ h) \to (k, P^ h)$ is the desired object of $\mathcal{D}\! \mathit{ef}_{P^ h}$. $\square$
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