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
\mathcal{D}\! \mathit{ef}_ P \longrightarrow \mathcal{D}\! \mathit{ef}_{S^{-1}P}
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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