Lemma 93.13.3. In Example 93.8.1 let $P$ be a $k$-algebra. Let $S \subset P$ be a multiplicative subset. If
$k \to P$ is of finite type, and
$\mathop{\mathrm{Spec}}(P) \to \mathop{\mathrm{Spec}}(k)$ is smooth at all points of $V(g)$ for all $g \in S$.
then the functor between deformation categories of Lemma 93.8.7
is smooth and induces an isomorphism on tangent spaces.
Proof. We know that $\mathcal{D}\! \mathit{ef}_ P$ and $\mathcal{D}\! \mathit{ef}_{S^{-1}P}$ are deformation categories by Lemma 93.8.2. Thus it suffices to check our functor identifies tangent spaces and a correspondence between liftability, see Formal Deformation Theory, Lemma 90.20.3. The property on liftability is proven in Lemma 93.13.1 and the isomorphism on tangent spaces is the special case of Lemma 93.13.2 where $N = B$. $\square$
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