Take the 2-minute tour ×  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 t