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Assume that $X,Y$ are affine schemes and $f,g\colon X\to Y$ morphisms between them. If the induced map $\mathrm{sp} (f),\mathrm{sp} (g)\colon\mathrm{sp} (X)\to\mathrm{sp} (Y)$ is the same(i.e. the induced map on the spectrum of these rings), are they the same morphisms? If not, are there any counter-examples?

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  • $\begingroup$ No. Let $X=\operatorname{Spec} \mathbb{F}_p[T]$, then both the maps $T \mapsto T^p$ and $T \mapsto T$ induce the identity map on the underlying topological spaces. $\endgroup$ Mar 18, 2020 at 14:48
  • $\begingroup$ Fixed, thanks for the suggestion. $\endgroup$ Mar 18, 2020 at 14:50

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No. Let $X=\operatorname{Spec} \mathbb{F}_p[T]$, then both the maps $T \mapsto T^p$ and $T \mapsto T$ induce the identity map on the underlying topological spaces.

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    $\begingroup$ Similarly: any field automorphism, e.g. complex conjugation on $\mathbb{C}$. $\endgroup$ Mar 18, 2020 at 23:03

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