# Syracuse Algebra Seminar

Fridays 3:30-4:25 PM, Carnegie 114

Organizers: Steven Diaz and Josh Pollitz

Schedule of Talks for the 2023-2024 Academic Year:

September 8th: Josh Pollitz (Syracuse University), Frobenius pushforwards and generators for the derived category

Abstract: By now it is quite classical that one can understand singularities in prime characteristic commutative algebra and algebraic geometry through properties of the Frobenius endomorphism. The foundational result illustrating this is a celebrated theorem of Kunz characterizing the regularity of a noetherian ring (in prime characteristic) in terms of whether a Frobenius push forward is flat. In this talk, I'll discuss a structural explanation of the theorem of Kunz, that also recovers it, and other theorems of this ilk. Namely, I’ll discuss recent joint work with Ballard, Iyengar, Lank, and Mukhopadhyay where we show that over an F-finite noetherian ring of prime characteristic high enough Frobenius push forwards generate the bounded derived category.

September 15th: Michael DeBellevue (Syracuse University), k-Summands of Syzygies of Burch Rings via the Bar Resolution

Abstract: In recent work, Dao and Eisenbud have defined the notion of a Burch ring. They have shown that for any module over a Burch ring of depth zero, the n'th syzygy contains direct summands of the residue field for all n>6. In joint work with Claudia Miller, we investigated how this behavior is explained by the bar resolution. I will define Burch rings and the bar resolution, and show how it can be used to describe some cycles which generate k-summands of syzygies of an R-module M. When M is a Golod module, we show that the number of these elements grows exponentially as the homological degree increases.

September 29th: Richard Bartels (Syracuse University), Generalized Loewy length of Cohen-Macaulay local and graded rings

Abstract:In his 1994 paper, Ding proved that for a Gorenstein local ring (R,m,k) with Cohen-Macaulay associated graded ring and infinite residue field, the generalized Loewy length gll(R) and index of R are equal. However, if k is finite, equality may not hold. In this talk, I will prove a relation between gll(R) and index(R) that generalizes Ding’s theorem for one-dimensional Cohen-Macaulay local rings with finite index. In particular, this generalization applies when the residue field is finite or infinite. I will then show that for certain one-dimensional hypersurfaces, this relation gives a formula for the generalized Loewy length. Finally, I will introduce a graded version of the generalized Loewy length, and determine its value for numerical semigroup rings.

October 6th: No seminar-Fall Break Weekend

October 13th: James Cameron (University of Utah)

October 20th: TBA

October 27th: Gordana Todorov (Northeastern University)

November 3rd: Dave Jorgensen (University of Texas-Arlington)

November 10th: Des Martin (Syracuse University)

November 17th: Torkil Stai (NTNU)

November 24th: No seminar-Thanksgiving Break

December 1st: Alexandra Seceleanu (University of Nebraska-Lincoln)

December 8th: David Lieberman (University of Nebraska-Lincoln)