Equivariance in Tannakian duality and plectic p-adic Hodge theory

Abstract

For a gerbe G and a finite constant group S acting on it, we use the Grothendieck construction to define a semidirect product gerbe G ⋊ S. We describe such gerbes in terms of Cech cocycles via the cohomology of crossed modules. We characterise a dual form of the Grothendieck construction and show that in the special case of the group S acting on a Tannakian category C, it gives rise to the category C^S of S-equivariant objects of C. If G and C correspond by Tannakian duality, then so do G ⋊ S and C^S. We derive descent for Tannakian categories from a classification of Galois gerbes and give a criterion for a descended category to be neutral. We then study a plectic variant of Fontaine theory for p-adic representations of the local plectic Galois group at p associated to a totally real number field F, when p is inert in F. In particular, we show that the plectic crystalline Fontaine functor is valued in a category of plectic isocrystals. We prove that this category is Tannakian using the theory developed in the first part.