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--$mathrm{HP}$ and crystalline--$mathrm{TP}$ spectral sequences need not degenerate.