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I've noticed that often authors will comment that "descent theory" shows that some sheaf in the étale topology is actually a scheme. I was wondering what result in descent theory actually implies this(a statement and reference for the result is what is requested in case an actual explanation would take too long). I understand how descent theory is what allows one to define and construct stacks, but I've never seen a result which says something to this effect (for example, I haven't seen it in Vistoli's stack notes in FGA explained) and it seems to be used often in verifying that something is a DM stack.

For example, Edidin states in the proof of Prop. 2.2 of his notes on $M_g$ that "$\underline{\mathrm{Iso}}_B(e,e')$ is the étale sheaf which is the quotient of $(X\times_{X\times X} E\times_B E')$ by the free group action of $G$. Moreover, descent theory shows that this sheaf is in fact a scheme."

In the proof of Theorem 3.2 there, he similarly writes, "Descent theory says that in this case a quotient $C=C_E/PGL(N+1)$ also exists as a scheme."

One finds similar statements in the other standard papers about DM stacks. I was wondering what results in "descent theory" they are referring to.

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    $\begingroup$ Minor note for HNuer and Chris Gerig: If by some chance it is difficult for you to type "é" using your keyboard, you could look up "etale topology" on Wikipedia (or your favorite search engine), and copy-paste from there. $\endgroup$
    – S. Carnahan ♦
    Commented Nov 21, 2012 at 5:16
  • $\begingroup$ You may want to look again in the last chapter of Vistoli's notes. He describes a few situations where descent is effective on schemes, and an example where it is not. Usually, the examples in the literature fall into one of the good situations, but unfortunately for readers, authors often don't bother to say precisely why. $\endgroup$
    – S. Carnahan ♦
    Commented Nov 21, 2012 at 5:26

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When you post a question, it would be good if you include enough explanations not to force the interested reader to go search for a paper online.

Anyway, the general question is: suppose that we have an fpqc covering of schemes $Y'\to Y$ and a scheme $X' \to Y'$ with descent data. When can I conclude that $X'$ descends to a scheme over $Y$? I know of two general results in this direction: this works when $X'$ is affine over $Y'$, of when there is a relative ample line bundle $L'$ on $X'$, and the descent data can be extended to $L'$. These are covered in my notes on descent theory: the first is in 4.3.1, the second in 4.3.3. Affine descent applies to the first of the cases you mention, descent via ample line bundles to the second.

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    $\begingroup$ I certainly apologize Angelo. In the future I will try to include more background to my question to make it self-contained. Thank you for pointing to those specific theorems in your notes. I of course had read those theorems when learning about descent, but the general use of the phrase "by descent theory" threw me off as to what statements were being used and how, as S. Carnahan pointed out above. Thanks for the response and sorry again. $\endgroup$
    – HNuer
    Commented Nov 21, 2012 at 14:54
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    $\begingroup$ You are welcome. There is no need to apologize. $\endgroup$
    – Angelo
    Commented Nov 21, 2012 at 15:42

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