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I've noticed that often authors will comment that "descent theory" shows that some sheaf in the $\acute{e}$tale 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.

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.

I've noticed that often authors will comment that "descent theory" shows that some sheaf in the '{e}tale 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.

I've noticed that often authors will comment that "descent theory" shows that some sheaf in the $\acute{e}$tale 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.