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

We construct the A-adic crystalline and Dieudonne analogues of Hida's ordinary A-adic etale cohomology, and employ integral p-adic Hodge theory to prove A-adic comparison isomorphisms between these cohomologies and the A-adic de Rham cohomology studied in Cais [The geometry of Hida families I: A-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida's A-adic etale cohomology. As applications of our work, we provide a 'cohomological' construction of the family of (phi, Gamma)-modules attached to Hida's ordinary A-adic etale cohomology by Dee [Phi-Gamma 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 A-adic duality theorems for each of the cohomologies we construct.