June 19, 2011

Specializing the fundamental group I

Posted by Akhil Mathew under algebraic geometry | Tags: , , , |
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(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 {X_0 \rightarrow \mathrm{Spec} k} be a smooth curve over an algebraically closed field {k} of characteristic {p}. We are interested in determining a set of topological generators for this curve. To do this, we started by showing that if {A} is a complete DVR with residue field {k}, then one can “lift” (by using cohomological vanishing and formal-to-algebraic comparison theorems) {X_0 \rightarrow \mathrm{Spec} k} to a smooth, proper morphism {X \rightarrow \mathrm{Spec} A}.

Ideally, we will have chosen {A} to be characteristic zero itself. Now our plan is to compare the two geometric fibers of {X}: one is {X_0}, and the other is {X_{\overline{\xi}}} (where {\xi} is the generic point; the over-line indicates that one wishes an algebraic closure of {k(\xi) = K(A)} here) with each other. Ultimately, we are going to show two things:

 

  1. The natural map {\pi_1(X_0) \rightarrow \pi_1(X)} is an isomorphism.
  2. The natural map {\pi_1(X_{\overline{\xi}}) \rightarrow \pi_1(X)} 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

\displaystyle \pi_1(X_{\overline{\xi}}) \rightarrow \pi_1(X_0).

However, {\pi_1(X_{\overline{\xi}})} will be seen to be topologically generated by {2g} generators (where {g} is the genus) by comparison with a curve over {\mathbb{C}}. For a smooth curve of genus {g} over {\mathbb{C}}, it is clear from the Riemann existence theorem (and the topological fundamental group) that {\pi_1} has {2g} topological generators.

Thus, it will follow, as stated earlier:

Theorem  If {X_0} is a smooth curve of genus {g} over an algebraically closed field of any characteristic, {\pi_1(X_0)} is topologically generated by {2g} generators.

 

One technical point will be, of course, that it is not entirely obvious that {\pi_1(X_{\overline{\xi}})} is the same as it would be for a curve over {\mathbb{C}}. This requires independent proof, but it will not be too hard.

(more…)

 

February 21, 2010

A few random things

Posted by Akhil Mathew under algebraic geometry, random | Tags: , , , |
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I don’t really anticipate doing all that much serious blogging for the next few weeks, but I might do a few posts like this one.

First, I learned from Qiaochu in a comment that the commutativity of the endomorphism monoid of the unital object in a monoidal category can be proved using the Eckmann-Hilton argument. Let 1 be this object; then we can define two operations on End(1) as follows.  The first is the tensor product: given a,b, define a.b := \phi^{-1} \circ a \otimes b \circ \phi, where \phi: 1 \to 1 \otimes 1 is the isomorphism.  Next, define a \ast b := a \circ b.  It follows that (a \ast b) . (c \ast d) = (a . c) \ast (b. d) by the axioms for a monoidal category (in particular, the ones about the unital object), so the Eckmann-Hilton argument that these two operations are the same and commutative. (more…)