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 $\mathcal D(\mathrm{ Frob}(\mathrm{ QCoh}(X),F_*)) \to \mathrm{ Frob}(\mathcal D(\mathrm{ QCoh}(X)),\mathcal D(F_*))$ of stable $\infty$-categories. Here, $\mathrm{ Frob}(-,-)$ denotes the $\infty$-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 $\infty$-category of Frobenius (and Cartier) modules satisfies Zariski descent.
Categories: math.AG math.AC