ScalingStacks

Proof. It suffices to show that $K$ preserves filtered colimits and split-exact sequences. The former follows from the fact that the $S^{\infty}_{\bullet}$ construction and restriction to the maximal subgroup preserve filtered colimits, as $\mathrm{N}(\Ar([n]))$ and $\Delta^{0}$ are compact $\infty$-categories. Corollary 7.9 allows us to reduce to consideration of split-exact sequences of spectral categories

$$\widehat{{\mathcal{A}}}_{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{C}}}_{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{B}}}_{\perf}.$$

As in [76], we observe that this sequence is Morita equivalent to the sequence

$$\widehat{{\mathcal{A}}}_{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}E(\widehat{{\mathcal{A}}}_{\perf},\widehat{{\mathcal{C}}}_{\perf},\widehat{{\mathcal{B}}}_{\perf})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{B}}}_{\perf}$$

(where $E$ denotes Waldhausen’s category of cofiber sequences in ${\mathcal{C}}$ with first term in the image of ${\mathcal{A}}$ and cofiber in the image of ${\mathcal{B}}$). Now Waldhausen’s additivity theorem implies the desired splitting on $K$-theory. ∎