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Consider a sort of $\mathbb{A}^1$-homotopy-stable algebraic $K$-theory for rings constructed as follows.

For $K_0$ we take a symmetrization subject to natural direct sum operation of $\mathbb{A}^1$-homotopy classes of finite-dimensional idempotents in $\mathbb{M}_\infty(B)$. Higher and lower groups are defined using looping/suspension. I hypothesize that this construction coincides with Weibel's homotopy algebraic K-theory $KH$, which is originally defined via spectra. This my be a trivial fact not even worth describing in the literature, or quite possibly I'm terribly wrong and missing some simple counterexample.

(Sorry if the question is too simple or too unclear - I haven't been doing real math for years.)

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    $\begingroup$ What kind of $\mathbb{A}^1$-homotopy are we talking about? Do you mean the $\mathbb{A}^1$-localization of the sheaf or are you just taking naive $\mathbb{A}^1$-classes? The second approach doesn't seem very likely to work out. And what do you mean by "suspension"? $\endgroup$ – Denis Nardin Jan 6 '20 at 12:06
  • $\begingroup$ It is the naive approach - my rings are in general not commutative, so there is no way to build sheaves. Suspension in this terms is a particular $\Sigma$ fitting into the short exact sequence $\Sigma A \to A[t, t^{-1}] \xrightarrow{ev_1} A$. $\endgroup$ – Kolya Ivankov Jan 6 '20 at 12:14
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    $\begingroup$ Thing is, I get this kind of theory as a byproduct of another construction, and it seems to be related to Cortinas-Thom bivariant algebraic theory $kk$, for which, given an $R$-aglebra $B$, one has $kk(E,B)\cong KH(B)$. I'm trying to prove the coincidence of my group with $KH_0$ via this sideway, but wonder if I'm inventing a bicycle by downgrading a jet fighter :) $\endgroup$ – Kolya Ivankov Jan 6 '20 at 12:19

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