(This is the third post in a project started here to describe some of Grothendieck’s work on the fundamental group of curves in positive characteristic.)

Let be a smooth curve over an algebraically closed field of characteristic . We are interested in determining a set of topological generators for this curve. To do this, we started by showing that if is a complete DVR with residue field , then one can “lift” (by using cohomological vanishing and formal-to-algebraic comparison theorems) to a smooth, proper morphism .

Ideally, we will have chosen to be characteristic zero itself. Now our plan is to compare the two geometric fibers of : one is , and the other is (where is the generic point; the over-line indicates that one wishes an algebraic closure of here) with each other. Ultimately, we are going to show two things:

- The natural map is an isomorphism.
- The natural map is an
*epimorphism.*

Here we have been loose with notation, as we have not indicated the relevant geometric points. The geometric point is, however, irrelevant for a connected scheme.

It will follow from this that there is an continuous epimorphism of profinite groups

However, will be seen to be topologically generated by generators (where is the genus) by comparison with a curve over . For a smooth curve of genus over , it is clear from the Riemann existence theorem (and the topological fundamental group) that has topological generators.

Thus, it will follow, as stated earlier:

TheoremIf is a smooth curve of genus over an algebraically closed field of any characteristic, is topologically generated by generators.

One technical point will be, of course, that it is not entirely obvious that is the same as it would be for a curve over . This requires independent proof, but it will not be too hard.