Commutative Algebra Reading Seminar

CARS Spring 2012

We meet 2:30-4pm on Tuesdays in Burnett 124. To receive our announcements, please visit the mailing list subscription page. You will need to register in order to send mail to the members of CARS.
CARS is currently organized by Amanda Croll and Courtney Gibbons, both of whom are conveniently located in Avery Hall 312.

Line up:

Updated Mon Feb 20 15:17:55 CST 2012

• February 21: Jason Lutz; Periodic free resolutions over complete intersections
  Abstract: We'll discuss a result of Eisenbud (1980) which states that a minimal resolution of a complete intersection by free modules of bounded rank is necessarily periodic of period 2.
• February 28: Becky Egg; On a paper of...
  Abstract: ...
• March 6: Luigi Ferraro; On a paper of...
  Abstract: ...
• March 13: Amanda Croll; On a paper of...
  Abstract: ...
• March 20: Haydee Lindo; On a paper of...
  Abstract: ...

Past talks:

• February 14: Kat Shultis; Rigidity of the Koszul complex
  Abstract: ...
• February 7: Michael Brown; More Matrix Factorizations
  Abstract: I will be speaking on a 2011 paper by Graham Leuschke and Ragnar-Olaf Buchweitz entitled "Factoring the Adjoint and Maximal Cohen-Macaulay Modules over the Generic Determinant". I will give a general overview of the paper and prove a key proposition which, according to the authors, is the basic link between factorizations of the adjoint of a generic matrix over a field and MCM modules.
• January 31: Brian Johnson; A Theorem of Gruson
  Abstract: According to Vasconcelos, this theorem of Gruson in some sense implies that "every finitely generated faithful module is 'piecewise' a generator for the category mod(R)." We'll discuss some of the main tools used in the proof, including a theorem of Serre relating to free summands of direct sums of projectives.
• January 24: Courtney Gibbons; From matrix factorizations to free resolutions
  Abstract: Last week we saw that matrix factorizations are awesome. This week we'll expand on that theme by constructing minimal free resolutions of MCMs over hypersurfaces. We'll end with a peek at another free resolution for arbitrary modules over a hypersurface. Notes (pdf)
• January 17: Jason Hardin; Cohen-Macaulay Modules on Hypersurface Singularities
  Abstract: A Theorem by Buchweitz, Greuel, and Schreyer states that a hypersurface of finite CM-representation type is a simple singularity. The proof of this result relies on the theory of matrix factorizations. We begin by discussing the basic theory of matrix factorizations and then use this to give a proof of the above theorem.

About CARS

We are a graduate student^* seminar that aims to complement the department's Commutative Algebra Seminar. The purpose of the Commutative Algebra Reading Seminar is threefold:

Grad students choose the content.

Traditionally, faculty do not attend CARS, and thus we get to decide what papers we want to read independent of direct faculty input. Of course, if your advisors suggests you read a paper, you are welcome to suggest that paper to the CARS participants!

Grad students get some practice reading papers and presenting in seminar.

One day, your advisor will inform you that you are speaking in Commutative Algebra Seminar; it's a good idea to have some talks under your belt before you present to the entire commutative algebra group.

Grad students in commutative algebra build a support network.

We're all learning a lot. It's helpful to keep on top of who knows what in the department!

^*Post-docs are encouraged to participate in CARS by claiming that they are graduate students who just happened to accidentally graduate. We like having post-docs participate in CARS!

KUMUNUjr

Embracing the spirit of KUMUNU, 2012's Compact CARS conference will be subsumed by KUMUNUjr. For more information about KUMUNUjr, contact Amanda Croll (NU), Courtney Gibbons (NU), Ananth Hariharan (NU), or Branden Stone (KU). (Or visit the website: KUMUNUjr 2012)

Compact CARS

Compact CARS is a miniconference that we run each year so that regular CARS participants have a chance to present their research to the usual suspects.

Spring 2011 Compact CARS:

• Some people spoke... (under construction)

Spring 2010 Compact CARS:

• Some people spoke...

Past Semesters

Fall 2012:

• December 6: Kat Shultis
• November 15: Luigi Ferraro
• November 1: Jason Lutz; On the invariant theory of finite groups
  Abstract: I'll talk about a paper by Müfit Sezer, "Sharpening the generalized Noether bound in the invariant theory of finite groups," (MR1933869) which proves that the degrees of the generators of the ring of invariants depend only on very basic properties of the group.
• October 25: Amanda Croll; Boij-Soederberg theory over a polynomial ring
  Abstract: I will build up the commutative algebra background necessary to understand Boij and Soederberg's conjecture on the decomposition of Betti diagrams of Cohen-Macaulay modules over a polynomial ring. This conjecture was proved by Eisenbud and Schreyer, and then extended by Boij and Soederberg.
• October 11: Special Event; Multiple presentations in preparation for the AMS sectional meeting.
• October 4: Zheng Yang; A Short Paper
  Abstract: I'll talk a little about a short paper.
• September 27: Ananth Hariharan; On a paper of ...
• September 20: Alexandra Seceleanu; Properties of Syzygies
  Abstract: I will discuss how to recognize a syzygy and will talk about some interesting properties they have.
• September 13: Courtney Gibbons; More on three-generated homology
  Abstract: Bruns's paper "'Jede' endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals" ('Every' finite free resolution is the free resolution of a three-generated ideal) generalizes some of the ideas that Brian Johnson discussed last month. (English translation available: arXiv:1012.5551v1, translated by Soumya Deepta Sanyal.)
• September 6: Michael Brown; On a paper of Eisenbud
  Abstract: I'll talk about the short paper "Subrings of Artinian and Noetherian Rings."
• August 30: Brian Johnson; On a paper of Burch
  Abstract: I'll present the paper "A note on the homology of ideals generated by three elements in local rings."
• August 23: Organizational Meeting
  Come with papers, ideas, requests, opinions...

Spring 2011:

• Hilbert sequences, functions, and series (oh my!)

Fall 2010:

Links

Papers:

These papers are still unclaimed. We decided we'd like to read these (Fall 2011):

• Tate, John. Homology of Noetherian rings and local rings
• Kunz, Ernst. Almost complete intersections are not Gorenstein rings
• Bongartz, Klaus. A generalization of a theorem of M. Auslander
• Eagon, John A.; Reiner, Victor. Resolutions of Stanley-Reisner rings and Alexander duality
• Miyata, Takehiko. Note on direct summands of modules
• Huneke, Craig; Leuschke, Graham J. On a conjecture of Auslander and Reiten
• Roberts, Paul. Abelian extensions of regular local rings
• Steinberg, Robert. Nagata's example
• Brodmann, M. Asymptotic stability of Ass(M/IⁿM)
• Swan, Richard G. The number of generators of a module

Commutative Algebra Resources:

• Recent commutative algebra papers on the arXiv
• Recent papers algebraic geometry papers on the arXiv
• Commalg.org: Database of upcoming conferences and events in commutative algebra
• Ravi Vakil's algebraic geometry course notes
• Commutative algebra questions on MathOverflow
• Algebraic geometry questions on MathOverflow