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 96.5.2. To see that it is true for $W$, we write $W = \mathop{\mathrm{colim}}\nolimits W_ n$ as in Formal Spaces, Lemma 85.16.9. 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 96.5.2). $\square$


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