The etale homotopy theory of a geometric fibration

Abstract

A geometric fibration, f:X→Y, is a smooth map of schemes which locally on Y admits a smooth, relative compactification. The etale homotopy type of the geometric fibre when completed away from the residue characteristics of Y,\((X_y )_{\hat et} \), is shown to be weakly homotopy equivalent to the completion of the Hurewicz fibre of the etale homotopy type of f,F(fetr)^. This implies a homotopy sequence for f. A key topological fact is verified in the appendix: for any pointed Hurewicz fibre triple F→E→B, the action of πl(F) on the homotopy type of various covering spaces of F extends to an action of πl(E).