# Factorizations of etale morphisms

Let $$f:X \rightarrow Y$$ be a finitely presented separated etale morphism, with $$Y$$ quasicompact and quasiseparated.

By Zariski’s main theorem, we can factor $$f$$ as $$f= g \circ j$$ with $$j$$ an open immersion and $$g$$ finite.

Can we choose $$g$$ to be flat?

• No, you cannot typically choose $g$ to be flat. Consider the morphism from an affine space of dimension $\geq 2$ to its quotient by a finite group of linear automorphisms that acts without pseudo-reflections. – Jason Starr Feb 14 at 3:14