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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 $A$?

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  • $\begingroup$ Re the beginning of question 1, there is no hope that the spectra are homotopy equivalent, because the units are already different, and the units appear as a direct summand of $K_1$. $\endgroup$
    – Steven Landsburg
    Commented Mar 5, 2014 at 16:35
  • $\begingroup$ Dear @FedeB, please do not ask questions simultaneously on MathOverflow and on Math.StackExchange. It leads to duplication of effort and is frowned upon by both communities. If you believe the question belongs on MathOverflow, please flag the question on Math.StackExchange for migration. $\endgroup$
    – Ricardo Andrade
    Commented Mar 5, 2014 at 17:02
  • $\begingroup$ @StevenLandsburg, how about $K_0$? $\endgroup$
    – FedeB
    Commented Mar 5, 2014 at 17:59
  • $\begingroup$ @FedeB: If the ring and the completion are both local, then of course the induced map on $K_0$ is an isomorphism. $\endgroup$
    – Steven Landsburg
    Commented Mar 5, 2014 at 18:08
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    $\begingroup$ Artin approximation provides very strong relations between henselian local rings and their completions. There is in fact a paper on K-theory and completion: G. Banaszak. An imbedding property for K-groups. JPAA 209 (2007), 239-244. It proves injectivity for the map on K-theory induced from the inclusion of a henselian local ring into its completion. It does not answer your question, but it may be a start. And of course, as noted above, it does not help with $K_0$-problems or anything more global. $\endgroup$
    – Matthias Wendt
    Commented May 17, 2014 at 9:33

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