Learning seminar on dualizing complexes, Fall 2026
Institute of Mathematics
Chinese Academy of Sciences
Emails: slyu [at] alumni [dot] princeton [dot] edu ; slyu [at] amss [dot] ac [dot] cn
We meet every Tuesday, 10--11 am, MCM 110. First meeting: Sept 1. Notes will be posted here.
No meeting on Sept 22. Assignment
No meeting on Sept 29 or Oct 5 (holidays).
No meeting on Nov 3 (likely).
Our plan: derived completion and local cohomology, local duality, dualizing complexes, (basic) discussion of several recent papers. Subsections do not necessarily corresponds to meetings. * = not planned/appeared in meetings, only appearing in notes. Derived Completion: Stacks Project, 091N,
0GLN,
0BKF,
0BKH.
Table of contents (tentative)
1. Derived completion and local cohomology [September]
1.1. Introduction and the Cech complex
1.2. Torsion and complete complexes
1.3. Universal properties and torsion-complete equivalence
*1.4. Complete modules and Nakayama
*1.5. Finite modules over Noetherian rings
*Appendix. The Koszul complex(es)
2. Complete local rings
2.1. Matlis duality [Oct 13]
2.2. Grothendieck local duality [Oct 13]
2.3. Standard properties of the normalized dualizing complex
2.4. Cohen--Macaulay and Gorenstein rings
3. Dualizing complexes
3.1 (Pseudo-)dualizing complexes
3.2 Upper shriek for (quasi-)projective morphisms
3.3 The !-tensor product
4. Complete (non-local) rings
4.1. !-lifting and !-descent
4.2. Ubiquity of (pseudo-)dualizing complexes
4.3. Generic injective hull and filterability of local cohomology
4.4. F-singularities of excellent rings
*5. Projective schemes over complete rings
References
Local Cohomology: Stacks Project, 0952,
0BJD.
Dualizing Complexes: Stacks Project,
08XG.
Papers to discuss: arXiv, 2604.20005,
2311.18196,
2503.13846 (appendix).
Notes: Oct 13.
Recordings: Google Drive. The meeting on Sept 8 was not recorded, and the other two recordings in September are of low quality.
Last updated: October 2026.