In this section we show that disjoint unions of spectra of Artinian rings can be descended along a quasi-compact surjective flat morphism of schemes.
Lemma 85.31.1. Let $X \to S$ be a morphism of schemes. Suppose $Y \to (X/S)_\bullet $ is a cartesian morphism of simplicial schemes. For $y \in Y_0$ a point define as a subset of $Y_0$. Then $y \in T_ y$ and $T_ y \cap T_{y'} \not= \emptyset \Rightarrow T_ y = T_{y'}$.
\[ T_ y = \{ y' \in Y_0 \mid \exists \ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\} \]
Proof. Combine Lemma 85.30.1 and Groupoids, Lemma 39.3.4. $\square$
Lemma 85.31.2. Let $X \to S$ be a morphism of schemes. Suppose $Y \to (X/S)_\bullet $ is a cartesian morphism of simplicial schemes. Let $y \in Y_0$ be a point. If $X \to S$ is quasi-compact, then is a quasi-compact subset of $Y_0$.
\[ T_ y = \{ y' \in Y_0 \mid \exists \ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\} \]
Proof. Let $F_ y$ be the scheme theoretic fibre of $d^1_1 : Y_1 \to Y_0$ at $y$. Then we see that $T_ y$ is the image of the morphism
Note that $F_ y$ is quasi-compact. This proves the lemma. $\square$
Lemma 85.31.3. 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 85.27.5. 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 85.31.1 above. By Lemmas 85.31.1 and 85.31.2 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. $\square$
To be sure, the lemma above has very limited applicability!
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