Lately I've looked at $f$-adic completions of commutative rings. I had posted two questions regarding the topic on math.SE which didn't receive any attention and I think they might be fit for mathoverflow. If not so, please feel free to tell me.
1.) Let $A$ a commutative ring, $f \in A$ a non zerodivisor. Let $\hat{A}$ the $f$-adic completion of $A$. Let $G$ be an $\hat{A}$-Module without $f$-torsion.
How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?
I have already shown $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$
Of course (as in my linked question), it holds again that $\varinjlim G/f^n G = G_f/G$, therefore it would suffice to show $\operatorname{Tor}_1^A(\hat{A}, G/f^n G) =0$ for all $n \geq 0$.
2.) Let $A$ a commutative ring, $f \in A$ not a zerodivisor and $\hat{A}$ the $f$-adic completion of $A$ (i.e. the completion along the ideal $(f)$).
For $A$ noetherian it is well-known that $\hat{A}$ is flat as an $A$-algebra. If $A$ is not noetherian however, this is not true in general.
Are there (explicit or general) examples where $A$ is not noetherian but its $f$-adic completion $\hat{A}$ is a flat $A$-algebra?