# Deligne–Mumford stack

• (i) the diagonal ${\displaystyle F\to F\times _{S}F}$ is representable (the base change to a scheme is a scheme), quasi-compact and separated.
A key fact about a Deligne–Mumford stack F is that any X in ${\displaystyle F(B)}$, B quasi-compact, has only finitely many automorphisms.