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Unformatted text preview: THE BAR CONSTRUCTION AND AFFINE STACKS MARTIN OLSSON 1. Introduction The purpose of this note is to clarify a technical point which arose in our work on nonabelian p-adic Hodge theory in [Ol]. While the main content of this paper is purely algebraic and not concerned with p-adic Hodge theory, let us first recall the motivation. 1.1. Let V be a complete discrete valuation ring with perfect residue field of characteristic p > 0 and fraction field K of characteristic 0. Fix an algebraic closure K ⊂ K . Let X/V be a smooth proper scheme, and let D ⊂ X be a divisor with normal crossings over V . Assume that X K is connected. Let X ◦ denote X- D and assume given a point x ∈ X ◦ ( V ). We can then consider the pro-unipotent fundamental group of ( X ◦ ,x ) in the various motivic realizations (Betti, ´ etale, de Rham, and crystalline). The ´ etale realization π et 1 ( X ◦ K ,x ) is the pro-unipotent group scheme over Q p which is Tannaka dual to the category of unipotent Q p- local systems on the geometric generic fiber X ◦ K . There is a natural action of the Galois group G K := Gal( K/K ) on π et 1 ( X ◦ K ,x ), and it is shown in [Ol] that the coordinate ring O π et 1 ( X ◦ K ,x ) is an ind-crystalline representation (that is, a direct limit of crystalline representations). Let us review what this means (see [Fo] for more details). Let W denote the ring of Witt vectors of the residue field of V , and let K ⊂ K denote the field of fractions of W . Let σ : W → W be the canonical lifting of Frobenius, and write also σ for the induced automorphism of K . Let B cris ( V ) denote Fontaine’s K-algebra defined for example in [Fo]. Recall that B cris ( V ) comes equipped with an action of G K , and a semi-linear automorphism ϕ . The ring B cris ( V ) is contained in a K-algebra B dR ( V ) which is a discrete valuation field. In particular, the valuation on B dR ( V ) defines a filtration on B dR ( V ). For a continuous finite dimensional G K-representation M over Q p , define D ( M ) := ( M ⊗ Q p B cris ( V )) G K , where the action of G K on M ⊗ Q p B cris ( V ) is the diagonal action. Then D ( M ) is a finite dimensional K-vector space with a semilinear automorphism induced by the automorphism of B cris ( V ). There is a natural map α M : D ( M ) ⊗ K B cris ( V ) → M ⊗ Q p B cris ( V ) which is always injective. The representation M is called crystalline if α M is an isomorphism. In this case we obtain an isomorphism D ( M ) ⊗ K B dR ( V ) ' M ⊗ Q p B dR ( V ) , which induces a filtration on D ( M ) ⊗ K K . 1 2 MARTIN OLSSON 1.2. It is shown in [Ol] that there is a canonical isomorphism of Hopf-algebras over K D ( O π et 1 ( X ◦ K ,x ) ) ' O π crys 1 ( X ◦ ,x ) , where π crys 1 ( X ◦ ,x ) denotes the Tannaka dual of the category of unipotent log isocrystals on ( X,D ) /W . This isomorphism is compatible with the Frobenius automorphisms, where the Frobenius automorphism on π crys 1 ( X ◦ ,x ) is induced by Frobenius pullback on the category of...
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