Local Cohomology, Spring 2018
Professor A.J. de Jong,
Department of Mathematics.
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:
- derived completion
- local cohomology
- complete vs torsion
- dualizing complexes
- local duality theorem
- annihilators of local cohomology
- 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.
- This semester I will give the lectures myself,
so there is no need to sign up for lectures.
- Please email me if you want to be on the associated mailing list.
- Time and place: Room 407, Fridays 10:30 -- 12:00 AM.
- First meeting: January 19. Everybody interested please attend.
Related discussion in Stacks project:
- Derived completion
- Local Cohomology
- Complete vs Torsion
- Dualizing Complexes
- Local duality
- Annihilators of local cohomology
- Finiteness theorem
- Completion of local cohomology modules and ML
(will be in lecture Friday 2-16)
- Algebraization of local cohomology
0EFP second (local)
- Algebraization of formal sections
- Algebraization of coherent formal modules
- Discussion of applications and further results.
- Grothendieck, SGA 2
- Mme Mich\`ele Raynaud, Theoremes de Lefschetz en cohomologie coherente et en cohomologie etale, Memoire de la SMF
- Faltings, A contribution to the theory of formal meromorphic functions,
Nagoya Math J
- Faltings, Der Endlichkeitssatz in der localen Kohomologie, Math Ann
- Faltings, Algebraization of Some Formal Vectorbundles, Annals of Math
- Faltings, Uber die Annulatoren lokaler Kohomologiegruppen, Arch Math
- Faltings, Erganzungen zu einem Artikel uber dualizierende Komplexe, Arch Math
- Faltings, Some Theorems about Formal Functions, Publ RIMS
- Faltings, Uber lokale Kohomologiegruppen hoher Ordnung, Journal fur die reine und angewandte Mathematik
- Faltings, Zur Existenz dualisierender Komplexe, Math Z