Appears in collection : Franco-Asian Summer School on Arithmetic Geometry

Similarly to crystalline cohomology theory, we give a local description of a prismatic crystal and its cohomology in terms of a q-Higgs module and its q-Dolbeault complex on a bounded prismatic envelope when the base prism is defined over the prism $Z_p[[q-1]]$. As an application, we obtain a comparison isomorphism between $A_{inf}$-cohomology of a relative Breuil-Kisin module, studied in a joint work with M. Morrow, and prismatic cohomology of the corresponding prismatic crystal. The result also can be applied to an analogous comparison in the integral p-adic Simpson correspondence.