Intersection Theory, Motives, Weil Cohomology Theories, de Rham Cohomology, Fall 2019

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

This semester I will teach this seminar as a course on the topics listed in the title.


  1. We will fix a base field k and work with separated schemes of finite type schemes over k usually denoted X, Y, Z.
  2. Define the Chow groups CH_*(X)
  3. Define proper pushforward on CH_*
  4. Define flat pullback on CH_*
  5. Define i^! for i the inclusion of an effective Cartier divisor
  6. Define the bivariant Chow groups A^*(X)
  7. Define the first chern class of an invertible module as an element of A^1(X)
  8. Prove the projective space bundle formula for CH_*
  9. Define c_i(-) and ch_i() for vector bundles
  10. Extend c(-) and ch(-) to K_0(Vect(X))
  11. Discuss K_0(Vect(X)) = K_0'(Coh(X)) = K_0(D^b_{Coh}) = K_0(D_{perf}(X)) for X smooth
  12. Prove ch gives isomorphsm K_0 ⊗ Q = CH_* ⊗ Q for X smooth
  13. Discuss GRR
  14. Define intersection product on CH_* for X smooth
  15. Define f^! on CH_* ⊗ Q for maps between smooth
  16. Discuss symmetric monoidal cats
  17. Introduce the category of correspondences
  18. Introduce the category of Chow motives
  19. Introduce classical Weil cohomology theory
  20. Give variant using c_1
  21. Introduce de Rham cohomology
  22. de Rham cohomology is a Weil cohomology theory


  1. This semester I will give the lectures myself.
  2. Please email me if you want to be on the associated mailing list.
  3. Time and place: Room 407, Fridays 10:30 -- 12:00 AM.
  4. First meeting: Friday, September 6 at 10:30 AM in Room 407.
  5. Rest of the schedule: September 13, NOT September 20 (AGNES), September 27, October 4, October 11, October 18, October 25, November 1, November 8, November 15, November 22, NOT November 29 (Thanksgiving), December 6.


[F] Intersection theory by Fulton

[SP-chow] Chow Homology and chern classes, Tag 02P3

[SP-weil] Weil Cohomology Theories, Tag 0FFG

[SP-de Rham] de Rham Cohomology Tag 0FK4

[G] La theorie des classes de Chern by Grothendieck

[K-cycles] Algebraic cycles and the Weil conjectures by Kleiman

[K-motives] Motives by Kleiman

[K-standard] The standard conjectures by Kleiman

[S] Classical Motives by Scholl