ScalingStacks

Proof. The localization theorem of [10, 7.1] implies $\{TC^{n}({\mathcal{C}})\}$ takes strict-exact sequences of spectral categories to strict exact sequences of small stable $\infty$-categories. Thus, we need to show that $TR^{n}({\mathcal{C}})$ preserve filtered colimits. We know this for $THH$, and the result now follows inductively from consideration of the fundamental cofibration sequence (e.g., [39, 2.1.4])

$$THH({\mathcal{C}})_{C_{p^{n-1}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TR^{n}({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}TR^{n-1}({\mathcal{C}})$$

(where the left-hand term denotes the homotopy orbit space) and the fact that homotopy orbits commute with filtered colimits. ∎