# Zariski's main for semi-separated targets

Quoting from Wikipedia:

If $$Y$$ is a quasi-compact separated scheme and $$f:X\to Y$$ is a separated, quasi-finite, finitely presented morphism then there is a factorization into $$X\to Z\to Y$$, where the first map is an open immersion and the second one is finite.

What is the strongest statement one can make if $$Y$$ is only assumed to be semi-separated, i.e. $$Y$$ has an affine diagonal (but $$f$$ is still a separated morphism)?

• You never trust Wikipedia. There is literally a stronger statement (no finite presentation assumption) for $Y$ having merely quasi-compact diagonal: stacks.math.columbia.edu/tag/05K0 – user138661 May 1 at 12:55