Prismatic crystals and q-Higgs fields

Similarly to the theory of crystalline cohomology, we give a local description of a prismatic crystal and its cohomology in terms of a q-Higgs module and the associated q-Higgs complex on the bounded prismatic envelope of an embedding into a framed smooth algebra, when the base bounded prism is defined over the q-crystalline prism. We also discuss its behavior under the scalar extension by the Frobenius lifting and tensor products, and a global description of the cohomology of a prismatic crystal via Zariski cohomological descent.