Algebraicity in monochromatic homotopy theory

Using Patchkoria–Pstrągowski’s version of Franke’s algebraicity theorem, we prove that the category of $K_p(n)$-local spectra is exotically equivalent to the category of derived $I_n$-complete periodic comodules over the Adams Hopf algebroid $(E_{\ast },E_{\ast }E)$ for large primes. This gives a finite prime result analogous to the asymptotic algebraicity for ${\mathrm{S}\mathrm{p}}_{K_p(n)}$ of Barthel–Schlank–Stapleton.