Local Cohomology, Spring 2018

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

Introduction. Grothendieck in SGA2 found a marvellous approach to coherent cohomology and Lefschetz theorems using dualizing complexes. We will revisit some of these results mostly in a purely algebraic setting. Here are some topics we will discuss:

  1. derived completion
  2. local cohomology
  3. complete vs torsion
  4. dualizing complexes
  5. local duality theorem
  6. annihilators of local cohomology
  7. finiteness theorem
Once these basic tools are in place our goal is to discuss Lefschetz theorems for local cohomology in commutative algebra. (It is up to the participants to force me to say how one deduces Lefschetz theorems in cohomology of projective varieties.) The form this type of Lefschetz theorem takes is that given ideals I, J of a Noetherian ring A and a finite module M one wants to relate limn H^i_J(M/I^nM) to the I-adic completion of H^i_J(M). One goal of the seminar is for the audience to help me get the sharpest possible result using the techniques above. In the literature the sharpest result appears to be due to Mme Raynaud, see reference given below.


  1. This semester I will give the lectures myself, so there is no need to sign up for lectures.
  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: January 19. Everybody interested please attend.

Related discussion in Stacks project:

  1. Derived completion 091N 0BKF 0BKH (Lecture 1)
  2. Local Cohomology 0BJA 0952 0BJD (Lecture 2)
  3. Complete vs Torsion 0A6V (Lecture 2)
  4. Dualizing Complexes 0A7A 0A7M 0A7W (Lecture 3)
  5. Local duality 0A81 (Lecture 3)
  6. Annihilators of local cohomology 0EFB (Lecture 4)
  7. Finiteness theorem 0AW7 0BJQ (Lecture 4)
  8. Completion of local cohomology modules and ML 0EEW 0EEX 0EFN (will be in lecture Friday 2-16)
  9. Algebraization of local cohomology 0EFF first 0EFP second (local) 0EFT bootstrap
  10. Algebraization of formal sections 0DXH 0EG1
  11. Algebraization of coherent formal modules 0DXS
  12. Discussion of applications and further results.


  1. Grothendieck, SGA 2
  2. Mme Mich\`ele Raynaud, Theoremes de Lefschetz en cohomologie coherente et en cohomologie etale, Memoire de la SMF
  3. Faltings, A contribution to the theory of formal meromorphic functions, Nagoya Math J
  4. Faltings, Der Endlichkeitssatz in der localen Kohomologie, Math Ann
  5. Faltings, Algebraization of Some Formal Vectorbundles, Annals of Math
  6. Faltings, Uber die Annulatoren lokaler Kohomologiegruppen, Arch Math
  7. Faltings, Erganzungen zu einem Artikel uber dualizierende Komplexe, Arch Math
  8. Faltings, Some Theorems about Formal Functions, Publ RIMS
  9. Faltings, Uber lokale Kohomologiegruppen hoher Ordnung, Journal fur die reine und angewandte Mathematik
  10. Faltings, Zur Existenz dualisierender Komplexe, Math Z