Let $R$ be a complete local integral domain with fraction field $K= \operatorname{Frac}(R)$ and residue field $k=R/m$. Let
$$ 0 \to A_1 \to A \to A_2 \to 0 $$
a short exact sequence of finite, flat $R$-algebras. Assume that this sequence split as algebras after tensoring by $K$ and $k$. That means that for $T=F, k$ there exist a section, ie a morphism $s_T: A_2 \otimes_R T \to A \otimes_R T$ of $T$-algebras which induces ismorphisms of $T$-algebras $f_T: A_1 \otimes_R A_2 \otimes_R T \to A \otimes_R T$ compatible with the sequence.
And the question is if this data suffice to assure that the sequence above splits; ie that there exist section $s: A_2 \to A$ which induces isomorphism $f: A_1 \otimes_R A_2 \to A$ in compatible manner.
A natural choice for section $s$ is the composition $A_2 \hookrightarrow A_2 \otimes_R K \to A \otimes_R K$ und the hope is that lands exactly in $A \hookrightarrow A \otimes_R K$. Note that $A_2 \hookrightarrow A_2 \otimes_R K$ is injective is governed by our assumption on flatness. Moreover $s_k: A_2 \otimes_R k \to A \otimes_R k$ implies that modulo quotient by maximal ideal in $R$ the image of $A_2$ indeed lands in $A$ und this would induce the map $f:A_1 \otimes_R A_2 \to A$.
The injectivity follows from injectivity of $f_K$, surjectivity should follow from $f_k$.
The point where I'm still not very convinced if these considerations really suffice to assure that $A_2 \hookrightarrow A_2 \otimes_R K \to A \otimes_R K$ lands in $A$. Also, can this problem be approached more conceptionally, eg analyzing $\operatorname{Ext}^1_R(A_2, A_1)$ groups, which as well known encode the information on extension classes of $A_1$ by $A_1$.
For example are there any canonical comparison maps relating $\operatorname{Ext}^1_R(A_2, A_1)$ to $\operatorname{Ext}^1_k(A_2 \otimes_R k, A_1 \otimes_R k)$, $\operatorname{Ext}^1_F(A_2 \otimes_R F, A_1 \otimes_R F)$ known, which could help to tackle this problem.
The described problem is related to an argument from Arithmetic Geometry, p 68. There originally the $A_i$ are Hopf-algebras und there it was indispensable to assume this to establish that $f_k$ is isomorphism.
But what if we assume now from the beginning that $f_K$ and $f_k$ are isomorphisms and our goal is to conclude from these assumptions that the sequence above split over $R$ or equivalently that $f$ is an isomorphism.
Should we still impose the Hopf algebra structure on $A_i$ and $A$, or is it at this stage not neccessary anymore and we obtain the desired splitting with pure homological algebra methods?