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

The goal of this note is to prove that Hodge-de Rham degeneration holds for smooth and proper ${\mathbf{F}}_p$-schemes $X$ with $\dim (X)<p^n$ as soon as its category of quasicoherent sheaves admits a lift to the truncated Brown-Peterson spectrum $\mathrm{B}\mathrm{P}⟨n-1⟩$, and the Hochschild-Kostant-Rosenberg spectral sequence for $X$ degenerates at the $E_2$-page. This is obtained from a noncommutative version, whose proof is essentially the same as Mathew’s argument in arXiv:1710.09045.