Counterexamples to Hochschild--Kostant--Rosenberg in characteristic $p$

We give counterexamples to the degeneration of the HKR spectral sequence in characteristic $p$, both in the untwisted and twisted settings. We also prove that the de Rham–$\mathrm{H}\mathrm{P}$ and crystalline–$\mathrm{T}\mathrm{P}$ spectral sequences need not degenerate.