I have this question just out of curiosity.


If X is a scheme, then a morphism $f: X \rightarrow X$ can be the identity on the underlying topological space of X, but not the identity on the structure sheaf. For example, f can be the Frobenius morphism.


Does someone know an example of such a morphism which is not a Frobenius? One can simply think in terms of affine schemes and hence ring: does someone know an explicit example of a ring endomorphism (sending 1 to 1) $\phi: X \rightarrow X$ such that $\phi^{-1}(I) = I$ for all prime ideals $I$ ? Is there a nice class of rings for which such an endomorphism is forced to be Frobenius?



Identity on topological space but not on scheme
I'm quoting an answer to close this question: 


There is the composition $k[ϵ]/ϵ^2→k→k[ϵ]/ϵ^2$, where the first arrow is the morphism of k-algebras sending $ϵ$ to 0 et the second one is the morphism making $k[ϵ]/ϵ^2$ into a $k$-algebra. – Damian Rössler


