Let $K$ be a number field and consider a arithmetic surface $X\to B=\operatorname{Spec} O_K$, i.e. $X$ is integral, regular, flat, proper over $O_K$ and it has dimension $2$.
Now pick a closed point $x\in X$ such that $x\mapsto b\in B$ and consider $\widehat{\mathcal O_{X,x}}$. In other words the completion of the local ring $\mathcal O_{X,x}$ with respect to its maximal ideal.
Can we express $\widehat{\mathcal O_{X,x}}= A[[t]]$? What is $A$ in this case? Do we have $A=O_L$ where $L$ is a complete discrete valuation field and a finite extension of $K_b$ (here $K_b$ is the completion of $K$ at $b$)?
For sure we have an embedding $\mathcal O_{B,b}[t]\hookrightarrow\mathcal O_{X,x}$
Thanks in advance