Let $X$ be an affine scheme. Let $f:X\rightarrow Y$ be an integral morphism that is an epimorphism in the category of schemes. Is $Y$ affine?
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$\begingroup$ No, take $X=\mathbf{A}^2 \setminus C$ where $C$ is a curve passing through $0$ which does not contain a line. Then we have a surjection $X\to \mathbf{P}^1$. $\endgroup$– Piotr AchingerCommented Apr 12, 2019 at 16:54
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$\begingroup$ See also: en.wikipedia.org/wiki/Jouanolou%27s_trick $\endgroup$– Piotr AchingerCommented Apr 12, 2019 at 16:54
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$\begingroup$ Or how about F = elliptic curve E less a pt and FxF -> E being group addition ? Won't that work too ? $\endgroup$– mehCommented Apr 12, 2019 at 18:32
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$\begingroup$ @aginensky yes, that works too. $\endgroup$– Piotr AchingerCommented Apr 12, 2019 at 20:00
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$\begingroup$ @PiotrAchinger is your morphism really integral? I kind of have hard time reconciling your claim with this: stacks.math.columbia.edu/tag/05YU $\endgroup$– user137767Commented Apr 13, 2019 at 0:53
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