The Geometry of Hida Families II: $\Lambda$-adic $(\phi ,\Gamma )$-modules and $\Lambda$-adic Hodge Theory

Abstract

We construct the $\Lambda$-adic crystalline and Dieudonné analogues of Hida's ordinary $\Lambda$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $\Lambda$-adic comparison isomorphisms between these cohomologies and the $\Lambda$-adic de Rham cohomology studied in the prequel to this paper as well as Hida's $\Lambda$-adic étale cohomology. As applications of our work, we provide a "cohomological" construction of the family of $(\phi ,\Gamma )$-modules attached to Hida's ordinary $\Lambda$-adic étale cohomology by the work of Dee, and we give a new and purely geometric proof of Hida's finitenes and control theorems. We also prove suitable $\Lambda$-adic duality theorems for each of the cohomologies we construct.

This paper is a continuation of our previous paper "The Geometry of Hida Families I: $\Lambda$-adic de Rham cohomology", and is a revised version of part of the paper arXiv:1209.0046

References in corpus (1)

• Dieudonne crystals and Wach modules for p-divisible fgroups

Cited by in corpus (1)

• Critical Lambda-adic modular forms and bi-ordinary complexes