Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration in characteristic $p>0$

Abstract The goal of this note is to prove that Hodge-de Rham degeneration holds for smooth and proper $\mathbf {F}_p$ -schemes X with $\dim (\text{X})<p^n$ assuming that two conditions hold: its category of quasicoherent sheaves admits a lift to the truncated Brown-Peterson spectrum $\mathrm {BP}\langle {n-1}\rangle $ , and the Hochschild-Kostant-Rosenberg spectral sequence for X degenerates at the $\mathrm{E}_2$ -page. This result is obtained from a noncommutative version thereof, whose proof is essentially the same as Mathew’s argument in [Mat20].