The derived $\mathrm{\infty }$-category of Frobenius modules

We prove that for $X$ a quasi-compact ${\mathbb{F}}_p$-scheme with affine diagonal (e.g.\ $X$ quasi-compact and separated) there is a t-exact equivalence $\mathcal{D}(\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}(\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}(X),F_{\ast }))\to \mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}(\mathcal{D}(\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}(X)),\mathcal{D}(F_{\ast }))$ of stable $\mathrm{\infty }$-categories. Here, $\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}(-,-)$ denotes the $\mathrm{\infty }$-category of generalized Frobenius modules as introduced in arXiv:2410.17102. This generalizes our result from arXiv:2410.17102, where we proved the above for regular Noetherian ${\mathbb{F}}_p$-schemes. As a byproduct we prove that the derived $\mathrm{\infty }$-category of Frobenius (and Cartier) modules satisfies Zariski descent.