The geometry of Hida families II: Λ-adic (p, Γ)-modules and Λ-adic Hodge theory

We construct the -adic crystalline and Dieudonné analogues of Hida's ordinary -adic étale cohomology, and employ integral -adic Hodge theory to prove -adic comparison isomorphisms between these cohomologies and the -adic de Rham cohomology studied in Cais [The geometry of Hida families I: -adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida's -adic étale cohomology. As applications of our work, we provide a 'cohomological' construction of the family of -modules attached to Hida's ordinary -adic étale cohomology by Dee [ - modules for families of Galois representations, J. Algebra 235 (2001), 636-664], and we give a new and purely geometric proof of Hida's finiteness and control theorems. We also prove suitable -adic duality theorems for each of the cohomologies we construct.