ScalingStacks

Proof. Using the proof of proposition 9.51 and mimicking definition 9.6, we can define a spectrum

$${I\mspace{-6.mu}K}^{{}^{\prime}}(R):=\colim_{n}\Omega^{n}K((\Psi_{\tri}(\hat{\mu}^{n}R)).$$

The conclusion of proposition 9.51 along with diagram 9.52 (which implies compatibility of the structure maps) yields an equivalence ${I\mspace{-6.mu}K}^{{}^{\prime}}(R)\simeq I\mspace{-6.mu}K(R)$. By the argument for [70, 11.7], we see that we can compute the homotopy groups of ${I\mspace{-6.mu}K}^{{}^{\prime}}(R)$ using a fibrant model that is a spectrum with $n$th space given by the space

$$\Omega^{\infty}K(\Psi_{\perf}(\hat{\mu}^{n}R)).$$

Lastly, lemma 9.39 and lemma 9.50 implies that there is an equivalence

$$\Omega^{\infty}K(\Psi_{\perf}(\hat{\mu}^{n}R))\simeq K_{0}(\mu^{n}\pi_{0}R)\times BGL^{+}(\hat{\mu}^{n}R).$$

Therefore, for $n>1$, $\pi_{0}\Omega^{\infty}K(\Psi_{\perf}(\hat{\mu}^{n}R))$ is $K_{0}(\mu^{n}\pi_{0}R)=K_{-n}(\pi_{0}R)$. ∎