Let $R$ be a $\bar{k}$-algebra (of finite type or complete) reduced (and maybe integral, if needed), let $A$ be an $R$-algebra, finite as an $R$-module, reduced and connected and such that there exists a section $\mathrm{id}\_R \colon R \rightarrow A \rightarrow R$. Is it true that $R \cong A$? If $R$ is an algebraically closed field, this is true (the section is not even needed), it is also true if $A$ is integral, but what in general (certainly we need at least the sections, or integral extensions of rings come to mind)?


**Remark.** Note that we are not assuming that $R$ is normal.



When is a finite $R$-algebra isomorphic to $R$?
I think this is not correct, take $k=\bar k$, $R=k[t]$, $A=k[x,y]/(xy)$ and the map $R \to A, t \mapsto x+y$. Then $A$ is a finite $R$-algebra and there is a section $A \to R, x \mapsto t, y \mapsto 0$.


Geometrically you have two crossing lines $L\_1, L\_2$ projecting to a single line $L$, with a section given by the inclusion of $L$ as $L\_1$. 


