Let $Y$ be an affine scheme of Krull dimension 1. Let $X\rightarrow Y$ be a monomorphism in the category of schemes. If $X$ is connected, is $X$ necessarily affine? What if we assume that $Y$ is a Noetherian scheme?
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$\begingroup$ Well, there are counterexamples such as e.g. $Y=\mathbb{A}^1_{\mathbb{C}}$, $X$ the disjoint union of $\operatorname{Spec}\mathbb{C}$ over all closed points of $Y$. Maybe you want to add some extra conditions on $X$ to avoid this type of scenario. $\endgroup$– dhyCommented Apr 12, 2019 at 14:15
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