I've encountered this problem, where I know everything in site (no pun intended) to be locally of finite type over my ground field, but I really need quasi-compactness.

Say I have two morphisms $a: X \to Y$ and $b: Y \to X$ such that $ba = id$ and say I know thaht $Y$ is of finite type, can I say anything about $X$?

This question comes from a moduli problem. I'm trying to prove that X is of finite type by knowing that Y is of finite type and that any family for X actually comes from a family for Y. So boundedness of families of Y should imply boundedness for families of X.