Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 02/05/2011 for the course MATH 224b taught by Professor Grunbaum,f during the Spring '08 term at University of California, Berkeley.

Page1 / 27


This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online