This is the fourth in a series of posts started here intended to describe Grothendieck’s work on the fundamental group of smooth curves in positive characteristic. Past posts were devoted to showing that a smooth proper curve could always be “lifted” to characteristic zero, via a proper smooth map (with a complete discrete valuation ring) whose specific fiber is the initial curve . The promise was to use various methods of algebraic geometry to compare the fundamental groups of the “generic” and the “special” fibers. In the previous post, we handled the case of the special fiber, where we argued that the fundamental group of the special fiber was the same as that of the whole thing. In this post, we want to develop the technology to handle the general fiber.

Our next goal is to obtain a small analog of the classical long exact sequence of a fibration in homotopy theory. Namely, the smooth proper morphism constructed as earlier that lifts the curve will be the “fibration,” and we are going to take the (geometric) fiber over the generic point. Since the base has trivial (because is complete local and its residue field is algebraically closed), it will follow from this long exact sequence that

is a surjection. (more…)