Math 746 - Topics in Ring Theory (Spring 2016)


Other references

Some references for local cohomology:
  1. Srikanth Iyengar et. al, Twenty-Four Hours of Local Cohomology
  2. Appendix 1 of David Eisenbud, The Geometry of Syzygies


This will be adjusted as necessary.
Jan 20Chapter 8: Summary of dimension theory
Jan 22Section 10.0: Principal ideal theorem
Section 10.1: Systems of parameters
Jan 25Section 17.1: Koszul complexes of lengths 1 and 2
Jan 27Section 17.2: Koszul complexes in general
Jan 29Section 17.3: Building the Koszul complex from parts
Feb 1Finish 17.3
Feb 3Guest lecture: Arinkin
Feb 5Guest lecture: Arinkin
Feb 8Section 18.1: Depth
Feb 10Section 18.2: Cohen-Macaulay rings
Feb 12Finish 18.2
Feb 15Section 16.6: Jacobian criteria for regularity
Section 18.3: Serre's conditions (Sn)
Feb 17Section 18.4: Flatness and depth
Section 18.5: Examples
Feb 19Section 19.1: Projective dimension and minimal resolutions
Section 19.2: Hilbert syzygy theorem
Feb 22Section 19.3: Auslander-Buchsbaum formula
Feb 24Section 19.4: Stably free modules and factoriality of regular local rings
Feb 26Section 20.1: Uniqueness of free resolutions
Feb 29Section 20.2: Fitting ideals
Mar 2Section 20.3: What makes a complex exact?
Mar 4Section 20.4: Hilbert-Burch theorem
Mar 7Section 20.5: Castelnuovo-Mumford regularity
Mar 9Section 15.1, 15.2: Monomials and monomial orderings
Mar 11Section 15.8: Gröbner bases and flat families
Mar 14Section 15.9: Generic initial ideals
Mar 16Section 15.3, 15.4: Division algorithm, Buchberger's criterion
Mar 18Section 15.5: Schreyer's theorem
Spring break
Mar 28Local cohomology: definitions, Čech complex
Mar 30Local duality (polynomial rings)
Apr 1Mayer-Vietoris sequence, Local cohomology and depth, sheaf cohomology
Apr 4no class
Apr 6no class
Apr 8no class
Apr 11Local cohomology and Hilbert polynomials, regularity
Apr 1324, Appendix, Section 11.1: Injective modules, Bass numbers
Apr 1524, Section 11.2: Gorenstein rings
Apr 1824, Section 11.4: Local duality (Gorenstein rings)
Apr 2024, Section 11.5: Canonical modules
Apr 22Canonical modules continued
Apr 25Local and sheaf cohomology
Apr 2724, Chapter 9: Cohomological dimension
Apr 2924, Section 15.2: Connectedness theorems of Faltings, Fulton-Hansen
May 2Finish 15.2
May 4Eagon-Northcott complex
May 6Kempf collapsing, see notes