Multivariable $(\varphi, \Gamma)$-modules and products of Galois groups

Abstract

We show that the category of continuous representations of the dth direct power of the absolute Galois group of Q p on finite dimensional F p -vector spaces (resp. finitely generated Z p -modules, resp. finite dimensional Q p -vector spaces) is equivalent to the category of étale (ϕ, Γ)-modules over a d-variable Laurent-series ring over F p (resp. over Z p , resp. over Q p ).Moreover, for each element α ∈ ∆ we have the partial Frobenius ϕ α , and group Γ α ∼ = Gal(Q p (µ p ∞ )/Q p ) acting on the variable X α in the usual way and commuting with the other *

Cohomology and Overconvergence for Representations of Powers of Galois Groups

“…Moreover, D commutes with filtered direct limits since both the tensor product and taking H Qp,∆ -invariants do so. Therefore D is an exact functor into the category lim [37]. On the other hand, for an object D in…”

“…We also decided not to replace Q p by a finite extension (or even by |∆| distinct extensions) in G Qp,∆ . One reason for this is that the paper [37] only covers representations of G Qp,∆ . Further, group cohomology of finite index subgroups of G Qp,∆ can easily be computed via G Qp,∆ -cohomology using Shapiro's Lemma.…”

“…For the coefficient ring E ∆ we require the stronger assumption for the étale property that D comes from an étale (ϕ ∆ , Γ ∆ )-module over O E ∆ by inverting p. The main result of [37] is that…”

“…The case i = 1 is treated in Prop. 4.1 in [37] and the higher cohomology groups vanish for the same reason: using the notations therein E ′ ∆ is cohomologically trivial for all finite extensions E ′ α /E α (α ∈ ∆) as it is induced as an H ′ -module.…”

“…In recent work [36,37] of the second named author the relevance of p-adic representations of a direct power of the absolute Galois group Gal(Q p /Q p ) to the p-adic Langlands programme is pointed out. The main result of [37] is that for any finite set ∆ the category of continuous representations of the group G Qp,∆ := α∈∆ Gal(Q p /Q p ) over F p (resp. over Z p , resp.…”