Field of fractions of the completion of a DVR

Question

I recently thought of this when working with $p$-adics - let $A$ be a discrete valuation ring with maximal ideal $\mathfrak{m}$, field of fractions $K$, and $\hat{A}$ the $\mathfrak{m}$-adic completion of $A$. Is it always true that the field of fractions $\hat K$ is isomorphic to $\hat{A} \otimes_A K$? If so, how can I prove this; if not, when does it hold?

Answer 1

Yes this is true; since $A$ is a DVR you have $K=A[1/\pi]$ where $\pi$ is a uniformizer. Since $\pi$ is also a uniformizer of $\widehat A$ you get for similar reasoning $K'=\widehat A[1/\pi]$ where $K'$ is the field of fractions of $\widehat A$. Thus

$$\widehat A\otimes_A K=\widehat A\otimes_A A[1/\pi]=\widehat A[1/\pi]=K'$$