The Stacks project

83.31 An example case

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 83.31.1. 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

\[ T_ y = \{ y' \in Y_0 \mid \exists \ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\} \]

as a subset of $Y_0$. Then $y \in T_ y$ and $T_ y \cap T_{y'} \not= \emptyset \Rightarrow T_ y = T_{y'}$.

Proof. Combine Lemma 83.30.1 and Groupoids, Lemma 39.3.4. $\square$

Lemma 83.31.2. 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

\[ T_ y = \{ y' \in Y_0 \mid \exists \ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\} \]

is a quasi-compact subset of $Y_0$.

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

\[ \xymatrix{ F_ y \ar[r] \ar[d] & Y_1 \ar[r]^{d^1_0} \ar[d]^{d^1_1} & Y_0 \\ y \ar[r] & Y_0 & } \]

Note that $F_ y$ is quasi-compact. This proves the lemma. $\square$

Lemma 83.31.3. 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 83.27.5. 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 83.31.1 above. By Lemmas 83.31.1 and 83.31.2 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 35.34.1. $\square$

To be sure, the lemma above has very limited applicability!


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 024F. Beware of the difference between the letter 'O' and the digit '0'.