Seminar Local Cohomology (WS 2014/2015) (eKVV: 24104)
Wednesday 10-12 Uhr, in V4-119 (2 SWS)
starting at the 15th of October .
Organizer: Prof. Dr. Henning Krause
, Prof. Dr. Christopher Voll
, Fajar Yuliawan
: Local cohomology was invented by Grothendieck in the early 1960s. Roughly speaking, it is a cohomology theory for sheaves on a topological space, which gives local information with respect to a closed subset (by taking the derived functors of sections). There are various versions of this theory and many applications depending on the context (algebra, geometry, topology).
The seminar provides an introduction to this beautiful theory, based to a large extent on the volume "Twenty-four hours of local cohomology".
At the moment, we plan to cover the following lectures from the book Twenty-four hours of local cohomology.
I. Fundamental notions
II. More commutative algebra
- Oct 15: Lecture 6. Complexes from a sequence of ring elements (Fajar Yuliawan)
- Oct 22: Lecture 7. Local cohomology (Fajar Yuliawan)
- Oct 29: Lecture 8. Auslander-Buchsbaum formula and global dimension (Ögmundur Eiriksson)
III. Algebraic geometry
- Nov 5: Lecture 9. Depth and cohomological dimension (Chao Zhang)
- Nov 12: Lecture 10. Cohen-Macaulay rings (Cosima Aquilino)
- Nov 19: Lecture 11. Gorenstein rings (Rebecca Reischuk)
IV. Combinatorics applications
- Nov 26: Lecture 12. Connections with sheaf cohomology (Julia Sauter)
- Dec 3: Lecture 13. Projective varieties (Philipp Lampe)
- Dec 10: Lecture 11 and 18. Local duality (Greg Stevenson)
- Dec 17: Lecture 5. Gradings, filtrations, and Gröbner bases (Zhi-Wei Li)
- Jan 7: Lecture 16. Polyhedral applications (Philipp Lampe)
- Jan 14: Lecture 20. Local cohomology over semigroup rings (Markus Perling)
- Jan 28: Linear homogeneous Diophantine equations and related Hilbert series (Christopher Voll)
- Feb 4: Local cohomology in triangulated categories (Henning Krause)
The participants of the seminar are supposed to be familiar with basic concept from (commutative and homological) algebra, including the language of rings and modules.
[B] Bruns, W. Commutative algebra arising from the Anand-Dumir-Gupta conjectures. Commutative algebra and combinatorics, 4, 1-38, (2007).
[BH] Bruns, W., and Herzog, J. Cohen-Macaulay rings. Cambridge University Press, 1998.
[BIK] Benson, D., Iyengar, S., and Krause, H. A local-global principle for small triangulated categories. arXiv preprint arXiv:1305.1668.
[FI] Foxby, H. B., & Iyengar, S. Depth and amplitude for unbounded complexes. Commutative algebra (Grenoble/Lyon, 2001), 119–137. Contemp. Math, 331.
[H] Hartshorne, R. Local cohomology. A seminar given by A. Grothendieck. Vol. 41. Springer-Verlag, 1967.
[I] Iyengar, S., et al. Twenty-four hours of local cohomology. Vol. 87. American Mathematical Soc., 2007.
[S1] Stanley, R. P. Hilbert functions of graded algebras, Advances in Math. 28 (1978), 57-83.
[S2] Stanley, R. P. Linear diophantine equations and local cohomology, Inventiones Math. 68 (1982), 175-193.