The Bar Construction and Affine Stacks

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 XK 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, étale, de Rham, and crystalline). The étale realization π 1 (X ◦ K , x) is the pro-unipotent group scheme over Qp which is Tannaka dual to the category of unipotent Qplocal systems on the geometric generic fiber X◦ K . There is a natural action of the Galois group GK := Gal(K/K) on π et 1 (X ◦ K , x), and it is shown in [Ol] that the coordinate ring Oπet 1 (X◦ K ,x)