Lemma 93.8.7. In Example 93.8.1 let $P$ be a $k$-algebra. Let $S \subset P$ be a multiplicative subset. There is a natural functor
of deformation categories.
Proof. Given a deformation of $P$ we can take the localization of it to get a deformation of the localization; 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 $\mathcal{D}\! \mathit{ef}_ P$. Let $S_ Q \subset Q$ be the inverse image of $S$. Then Hence $(A, S_ Q^{-1}Q) \to (k, S^{-1}P)$ is the desired object of $\mathcal{D}\! \mathit{ef}_{S^{-1}P}$. $\square$
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