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.



How does descent theory imply a sheaf is a scheme?
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.


