Fundamental Groups of Schemes, Fall 2015

Professor A.J. de Jong, Columbia university, Department of Mathematics.

This semester I will teach a topics course in algebraic geometry whose topic will be Grothendieck's fundamental group and applications.

Time and place: Tuesday and Thursday, 1:10 - 2:25 PM in Room 507 (note change of room).

Material covered in lectures so far:

  1. Introduction using the case of topological spaces. Discussion of topological groups, profinite spaces, profinite groups. Topology on the automorphisms of a functor from a category to a set (Lemma Tag 0BMR.
  2. Galois categories: Section Tag 0BMQ up to Lemma Tag 0BN3.
  3. Finish the section on Galois categories. Stated basic properties of etale morphisms (see for example Section Tag 02GH and Chapter Tag 024J) and finite etale morphisms.
  4. Etale fundamental groups: Section Tag 0BQ8
  5. Etale fundamental groups of schemes locally of finite type over the complex numbers, see SGA I, Expose XII.
  6. Etale fundamental groups of normal schemes, see Section Tag 0BQJ, Section Tag 0BSN, and Section Tag 0BSD.
  7. Galois action on the geometric fundamental group, see Section Tag 0BTU.
  8. Ramification theory, see Section Tag 09E3 and Lemma Tag 0BSC. Also, topological invariance of the etale topology and etale fundamental group, see Section Tag 04DY and Section Tag 0BTT.
  9. Finite etale covers of proper schemes, see Section Tag 0BQC
  10. Purity of branch locus, see Section Tag 0BJE. We also very briefly discussed Proposition Tag 0BPD (was stated as a corollary in the lecture) but unfortunately, the theorem stated just before this was stated incorrectly. The notes below by Pak-Hin Lee have the correction in them. Please ask me if you are still confused after reading in these notes.
  11. Local Lefscetz for fundamental groups. We explained the proof of Lemma Tag 0BM6.
  12. Specialization of the fundamental group: proper smooth case. In this lecture we discussed LemmaTag 0A49 and the material of this commit, this commit, and this commit.
  13. Abhyankar's lemma and conclusion of the proof of the theorem on specialization of fundamental groups in the smooth proper case. A very general version of Abhyankar's lemma is Lemma Tag 0BRM which is discussed in the long section Section Tag 09EL which is about Epp's result on eliminating wild ramification.
  14. Quasi-unipotent monodromy theorem. We discussed what the theorem says over the complex numbers and then we stated Grothendieck's algebraic geometric version for smooth proper schemes over discretely valued fields. We also proved the first part, which says that the image of wild inertia is finite.
  15. Quasi-unipotent monodromy theorem when the residue field has finitely many l-power roots of 1. We explained why this is purely a result about the structure of the decomposition group in this case. As an example we discussed what happens with torsion points on abelian varieties over discretely valued fields. We also explained why the Lefschetz principle does not allow one to reduce the general case to this special case.
  16. General case of Grothendieck's quasi-unipotent monodromy theorem. This relies on Krasner's lemma and N\'eron desingularization as well as a generalization of Abhyankar's lemma which we stated in the lecture.
  17. Finish of the discussion of Grothendieck's quasi-unipotent monodromy theory. A reference for this result is SGA 7 (tome 1), Exposee 1 (by Deligne). Extra material: Lecture notes on Witt vectors by Lars Hesselhot. Other sources for Witt vectors are Serre's Local Fields and Hazewinkel's Formal Groups.

There are live-TeXed notes by Pak-Hin Lee.

Here is a list of literature to look at:

  1. SGA 1
  2. Murre, Lectures on an introduction to Grothendieck's theory of the fundamental group
  3. Stacks project chapter on fundamental groups
  4. Lenstra, Galois Theory for Schemes
  5. Grothendieck, Murre, The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme (Springer Link for Columbia)
  6. Lieblich, Olsson, Generators and Relations for the Etale Fundamental Group