Prismatic Crystals, q-Higgs Modules, and their Cohomology

We study prismatic crystals and their cohomology by using q-Higgs modules (= a q-analogue of p-connections). When the base is lying over the q-crystalline prism, they are locally described in terms of q-Higgs modules and the associated complexes on the bounded prismatic envelope in a smooth algebra. This allows us to understand the isogeny property of Frobenius and the Nygaard filtration in terms of q-Higgs modules. For a general base (R,I) with R/I containing a primitive p-th root, we obtain a similar description by adding the variable q and the q-Higgs derivation by q considered in a recent work by Gros, Le Stum, and Quirós. This also works in the absolute case for a ring containing a primitive p-th root.