I would be grateful if someone could explain to me the error in the following computation.
Assume that M is a topologically complete $\mathbb C[[t]]$-module. Let $K:=\mathbb C((t))$ be the field of formal Laurent series. By definition, the completed tensor product $M\hat{\otimes}_{\mathbb C[[t]]}K$ is given by
$$M\hat{\otimes}_{\mathbb C[[t]]}K=invlim_{p, q\geq0}\frac{M}{(t)^p}\otimes\frac{K}{(t)^q}.$$
Now, since $K$ is a field, it follows that $K/(t)^q=0$ for every integer $q\geq0$. Hence, the above implies
$$M\hat{\otimes}_{\mathbb C[[t]]}K=0.$$
This is clearly nonsense since in the special case $M=\mathbb C[[t]]$, we have $M\hat{\otimes}_{\mathbb C[[t]]}K=K$, which is not $0$. However, I am unable to identify the mistake in my computations. Any suggestion would be appreciated.