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?

  • 3
    $\begingroup$ 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. $\endgroup$ – Jason Starr Feb 14 at 3:14

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