On arithmetic families of filtered phi-modules and crystalline representations

Families of $p$-adic Galois representations and $(\phi, \Gamma)$-modules

“…It was already pointed out in our previous paper [He1] that the right category to handle these objects is the category of adic spaces (locally of finite type over Q p ) as introduced by Huber, see [Hu]. Using the language of adic spaces, we show in this paper that given a family N of (ϕ, Γ)-modules over the relative Robba ring B † X,rig on an adic space X locally of finite type over Q p (see below for the construction of the sheaf B † X,rig ), one can construct natural open subspaces X int resp.…”

“…In this section we define relative versions of the classical period rings used in the theory of (ϕ, Γ)-modules and in p-adic Hodge-theory. Some of these sheaves were already defined in [He1,8].…”

“…are the completions of the ordinary tensor product for the topology that is given by the discrete topology on A + /p i A + and by the natural topology on W i ( Ẽ+ ) resp. W i ( Ẽ), see [He1,8.1].…”

“…Let X be a reduced adic space locally of finite type over Q p . As in [He1,8.1] we can define sheaves Ẽ + X , ẼX , Ã + X and ÃX by demanding…”

“…Note that all the rational period rings (i.e. those period rings in which p is inverted) can also be defined on a non-reduced space X by locally embedding the space into a reduced space Y and restricting the corresponding period sheaf from Y to X, compare [He1,8.1].…”