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Is there a morphism in the category of schemes which is simultaneously a monomorphism and an epimorphism yet is not an isomorphism? "Nicer" examples are preferred (e.g. with integral Noetherian separated source and target).

There are such morphisms in the category of commutative unital rings e.g. $\mathbb{Z}\rightarrow \mathbb{Q}$ (https://math.stackexchange.com/a/2792241/690882, https://mathoverflow.net/q/109). An epimorphism in the category of commutative unital rings defines a monomorphism in the category of schemes (I think) but a monomorphism in the category of commutative unital rings is not necessarily an epimorphism in the category of schemes (https://math.stackexchange.com/a/3302479/690882).

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Sure, take a local domain $A$ and its field of fractions $Q = \operatorname{Quot} A$. Then $\operatorname{Spec} Q → \operatorname{Spec} A$ is epi and mono in the category of schemes, but no isomorphism.

This is because of the following:

  • As you said, an epimorphism in the category of rings indeed induces a momomorphism in the category of schemes. To see this, just note that equality of morphisms is local on the source.
  • Every morphism of schemes coming from a spectrum of a local ring factorizes through every open neighbourhood of its image of the unique closed point in the source.

So in both cases, we can reduce to the affine case.

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    $\begingroup$ I think for the second bullet point it may be relevant to mention EGA I Prop. 2.4.4 (old edition). $\endgroup$ – user690882 Jul 24 '19 at 15:01

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