Isomorphism between finite algebras over ${\Bbb Z}_p$

Question

Let $\pi \colon R \twoheadrightarrow {\Bbb T}$ be a surjective ring homomorphism between finite algebras over ${\Bbb Z}_p$. Further, we suppose the following three conditions$\colon$

1. $R$ is a complete intersection, i.e. $R = {\Bbb Z}_p[[X_1,\ldots,X_d]]/(f_1,\ldots,f_d)$.

2. $\pi$ induces an isomorphism $\pi^{*} \colon {\mathrm{Hom}}_{{\Bbb Z}_p}({\Bbb T}, \overline{{\Bbb Z}_p}) = {\mathrm{Hom}}_{{\Bbb Z}_p}(R, {\overline{\Bbb Z_p}})$, where we denote by $\overline{{\Bbb Z}_p}$ the integral closure of ${\Bbb Z}_p$ in the algebraic closure $\overline{{\Bbb Q}_p}$.

3. ${\Bbb T}$ is reduced.

We shall denote by $R^{\mathrm{red}}$ the reducification of $R$. Then, I would like to ask

Q. Do the above three conditions imply the isomorphism $\pi \colon R^{\mathrm{red}} \cong {\Bbb T}$?

In the case where $\overline{{\Bbb Z}_p}$ is replaced with ${\Bbb Z}_p$, the question was answered in the negative by Professor Spivakovsky.

Answer 1

The answer is ``no''. Let $R=\frac{\mathbb Z_p[X]}{(X^2-p)(X^3-p)}$ and $\mathbb T=\frac{\mathbb Z_p[X]}{(X^2-p)}$. The ring $R$ is a hypersurface, hence a complete interserction; in addition, it is reduced, so $R=R^{red}$. The ring $\mathbb T$ is also reduced. The map $\pi^*$ is the zero map between two modules, each of which is equal to $(0)$, so it is an isomorphism. Yet, $R=R^{red}$ is not isomorphic to $\mathbb T$.

Answer 2

The answer to the modified question (that is, with $\overline{\mathbb Z_p}$ instead of $\mathbb Z_p$) is "yes".

The prime $p$ is not a zero divisor in $R$. Replacing $R$ by $R^{red}$ does not change the problem (we no longer claim that $R$ is a complete intersection, but $p$ is still not a zero divisor). We will prove that $\pi$ is an isomorphism. Assume the contrary, aiming for contradiction. Let $I=Ker(\pi)$. Then there exists a minimal prime $P$ of $R$ such that $I\not\subset P$, so $\pi$ induces a map $\frac RP\rightarrow\frac R{I+P}$ with $\frac R{I+P}$ zero-dimensional.

The ring $\frac RP$ contains $\mathbb Z_p$ and is integral over it, hence admits a non-zero homomorphism to $\overline{\mathbb Z_p}$. This homomorphism does not factor through $\mathbb T$, contradicting the hypothesis 2.