Lemma 93.11.2. Example 93.11.1 satisfies the Rim-Schlessinger condition (RS). In particular, $\mathcal{D}\! \mathit{ef}_ X$ is a deformation category for any algebraic space $X$ over $k$.
Proof. Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda $. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma 90.16.4 it suffices to show that the functor $\mathcal{F}(A_1 \times _ A A_2) \to \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories. Observe that
\[ \xymatrix{ \mathop{\mathrm{Spec}}(A) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A_2) \ar[d] \\ \mathop{\mathrm{Spec}}(A_1) \ar[r] & \mathop{\mathrm{Spec}}(A_1 \times _ A A_2) } \]
is a pushout diagram as in Pushouts of Spaces, Lemma 81.6.2. Thus the lemma is a special case of Pushouts of Spaces, Lemma 81.6.7. $\square$
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