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The geometry of Hida families II:
$\unicode[STIX]{x1D6EC}$-adic 
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules and 
$\unicode[STIX]{x1D6EC}$-adic Hodge theory
- Part of
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- Journal:
- Compositio Mathematica / Volume 154 / Issue 4 / April 2018
- Published online by Cambridge University Press:
- 08 March 2018, pp. 719-760
- Print publication:
- April 2018
-
- Article
- Export citation
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We construct the
$\unicode[STIX]{x1D6EC}$-adic crystalline and Dieudonné analogues of Hida’s ordinary 
$\unicode[STIX]{x1D6EC}$-adic étale cohomology, and employ integral 
$p$-adic Hodge theory to prove 
$\unicode[STIX]{x1D6EC}$-adic comparison isomorphisms between these cohomologies and the 
$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology studied in Cais [The geometry of Hida families I:
$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s 
$\unicode[STIX]{x1D6EC}$-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of 
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules attached to Hida’s ordinary 
$\unicode[STIX]{x1D6EC}$-adic étale cohomology by Dee [
$\unicode[STIX]{x1D6F7}$–
$\unicode[STIX]{x1D6E4}$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 
$\unicode[STIX]{x1D6EC}$-adic duality theorems for each of the cohomologies we construct.
