The Stacks project

Lemma 84.31.3. Let $X \to S$ be a quasi-compact flat surjective morphism. Let $(V, \varphi )$ be a descent datum relative to $X \to S$. If $V$ is a disjoint union of spectra of Artinian rings, then $(V, \varphi )$ is effective.

Proof. Let $Y \to (X/S)_\bullet $ be the cartesian morphism of simplicial schemes corresponding to $(V, \varphi )$ by Lemma 84.27.5. Observe that $Y_0 = V$. Write $V = \coprod _{i \in I} \mathop{\mathrm{Spec}}(A_ i)$ with each $A_ i$ local Artinian. Moreover, let $v_ i \in V$ be the unique closed point of $\mathop{\mathrm{Spec}}(A_ i)$ for all $i \in I$. Write $i \sim j$ if and only if $v_ i \in T_{v_ j}$ with notation as in Lemma 84.31.1 above. By Lemmas 84.31.1 and 84.31.2 this is an equivalence relation with finite equivalence classes. Let $\overline{I} = I/\sim $. Then we can write $V = \coprod _{\overline{i} \in \overline{I}} V_{\overline{i}}$ with $V_{\overline{i}} = \coprod _{i \in \overline{i}} \mathop{\mathrm{Spec}}(A_ i)$. By construction we see that $\varphi : V \times _ S X \to X \times _ S V$ maps the open and closed subspaces $V_{\overline{i}} \times _ S X$ into the open and closed subspaces $X \times _ S V_{\overline{i}}$. In other words, we get descent data $(V_{\overline{i}}, \varphi _{\overline{i}})$, and $(V, \varphi )$ is the coproduct of them in the category of descent data. Since each of the $V_{\overline{i}}$ is a finite union of spectra of Artinian local rings the morphism $V_{\overline{i}} \to X$ is affine, see Morphisms, Lemma 29.11.13. Since $\{ X \to S\} $ is an fpqc covering we see that all the descent data $(V_{\overline{i}}, \varphi _{\overline{i}})$ are effective by Descent, Lemma 35.37.1. $\square$

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