I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community.

I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of Quillen) of a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ with the $K$-theory of the $\mathfrak{m}$-adic completion $\hat{A}$ of $A$.

**Question 1:** Is there any hope that the two spectra $K(A)$ and $K(\hat{A})$ are homotopy equivalent, which means: what should one ask to the ring $A$ to make this true? My mind connects some related statement to the name of Karoubi, but Google did not help.

**Edit:** as noticed in the comments, this is hopeless. I change my question in the following

**Question 1':** Let $B$ be a Noetherian ring, $J$ an ideal of $B$. Is it possible that $K_0(B) \cong K_0(\hat{B})$, where $\hat{B}$ denotes here the $J$-adic completion of $B$?

Back to the local world. A more precise (and possible less hopeless) related question. Let $I\subsetneq \mathfrak{m}$ be a principal ideal of $A$, generated by a non-zero divisor. Let $X=\mathrm{Spec}({A})$, $Y=\mathrm{Spec}({A/I})$. One has a canonical push forward map

$$i_{*}\colon K(Y)\to K(X)$$ giving the corresponding morphism on $K_0$ groups.

Consider an $A$-module $M$, finitely generated, of finite projective dimension (i.e. admitting a finite projective resolution). I want to understand when $[M]$ in $K_0(A)$ is in the image of the push forward map from $K_0(A/I)$, i.e. when there is a $K_0$ class $\alpha\in K_0(A/I)$ such that $i_{*}(\alpha) = [M]$.

**Question 2:** can I replace $A$ with $\hat{A}$ to answer to this kind of questions? Which extra condition should I add to $M$ in order to be able to descend the answer on $\hat{A}$ to an answer on $A$?