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The Stacks project

Proof. Recall that the condition only involves the evaluation F(V) of the functor F on schemes V over S which are spectra of Artinian local rings and the restriction maps F(V_2) \to F(V_1) for morphisms V_1 \to V_2 of schemes over S which are spectra of Artinian local rings. Thus let V/S be the spetruim of an Artinian local ring. If \xi = (Z, u', \hat x) \in F(V) then either Z = \emptyset or Z = V (set theoretically). In the first case we see that \hat x is a morphism from the empty formal algebraic space into W. In the second case we see that u' is a morphism from the empty scheme into X' and we see that \hat x : V \to W is a morphism into W. We conclude that

F(V) = U'(V) \amalg W(V)

and moreover for V_1 \to V_2 as above the induced map F(V_2) \to F(V_1) is compatible with this decomposition. Hence it suffices to prove that both U' and W satisfy the Rim-Schlessinger condition. For U' this follows from Lemma 98.5.2. To see that it is true for W, we write W = \mathop{\mathrm{colim}}\nolimits W_ n as in Formal Spaces, Lemma 87.20.11. Say V = \mathop{\mathrm{Spec}}(A) with (A, \mathfrak m) an Artinian local ring. Pick n \geq 1 such that \mathfrak m^ n = 0. Then we have W(V) = W_ n(V). Hence we see that the Rim-Schlessinger condition for W follows from the Rim-Schlessinger condition for W_ n for all n (which in turn follows from Lemma 98.5.2). \square


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