# Are morphisms between affine schemes the same?

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?

• 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. Mar 18, 2020 at 14:48
• Fixed, thanks for the suggestion. Mar 18, 2020 at 14:50

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