Unramified F-divided objects and the etale fundamental pro-groupoid in positive characteristic

摘要

Let X/S be a flat algebraic stack of finite presentation. We define a new etale fundamental pro-groupoid Π_1(X/S), generalizing Grothendieck's enlarged etale fundamental group from SGA 3 to the relative situation. When S is of equal positive characteristic p, we prove that Π_1(X/S) naturally arises as colimit of the system of relative Frobenius morphisms X → X~(p/S) → X~(p~2/S) → ... in the pro-category of Deligne Mumford stacks. We give an interpretation of this result as an adjunction between Π_1 and the stack Fdiv of F-divided objects. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic p.