# Syracuse Algebra Seminar

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

Organizers: Steven Diaz and Josh Pollitz

Schedule of Talks for Fall 2023:

This week's talk:

December 1st: Alexandra Seceleanu (University of Nebraska-Lincoln), Principal symmetric ideals

Abstract: A ubiquitous theme in mathematics is that general members in a family of mathematical objects have nice properties. We focus on the family of principal symmetric ideals, that is, ideals generated by the orbit of a homogeneous polynomial under the action of the symmetric group. In joint work with Megumi Harada and Liana Sega we determine the minimal free resolutions for a general member of this family, as well as other notable properties it satisfies.

Upcoming talk:

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

Past talks:

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), Koszul ring maps and transferring resolutions in local algebra

Abstract: I will discuss what it means for a map of commutative local rings to be Koszul, and how a Koszul ring map allows homological information to be transferred from the source to the target. This gives, in many cases, constructions of “universal resolutions” of modules over a non-regular local ring in terms of a finite amount of data. I will discuss these universal resolutions, and how the construction of these resolutions generalizes other transfer-of-resolutions results, such as the Eisenbud-Shamash construction. A key ingredient in these results are A-infinity algebra techniques, and I will give a brief introduction to A-infinity algebras and their role as a helpful generalization of dg-algebras. All this work is joint with Benjamin Briggs, Janina Letz, and Josh Pollitz.

October 27th: Gordana Todorov (Northeastern University), Higher Auslander Algebras

Abstract: A celebrated theorem of Maurice Auslander about artin algebras describes the correspondence between {algebras A of representation finite} and {algebras B, with gl.dim.B ≤ 2 ≤ dom.dim.B} which are now called Auslander algebras. Higher Auslander algebras were introduced by O. Iyama as algebras C, with {gl.dim.C ≤ k ≤ dom.dim.C} and it was also shown that there is correspondence between {higher representation finite algebras} and {higher Auslander algebras} (up to certain equivalences). A large part of my talk is on recent work of Emre Sen. In addition, Emre Sen, Shijie Zhu and I obtained another method of constructing higher Auslander algebras in the family of Nakayama algebras. Using this method we get a complete characterization of higher Auslander Nakayama algebras.

November 3rd: Dave Jorgensen (University of Texas-Arlington), Variations on a classical dimension inequality

Abstract: For a pair of finitely generated modules M and N over a codimension c complete intersection ring R with M⊗N having finite length over R, we pay special attention to the inequality dim(M)+dim(N) ≤ dim(R) +c. In particular, we develop an extension of Hochster's theta invariant whose nonvanishing detects equality. In addition, we consider a parallel theory where dimension and codimension are replaced by depth and complexity, respectively. This is joint work with Petter Bergh and Peder Thompson.

November 10th: Des Martin (Syracuse University), On the splitting of the 2nd exterior power of the conormal module from Tor

Abstract: Let R be a local Noetherian ring and I an ideal of R. In “The Syzygies of the Conormal Module,” Simis and Vasconcelos give an embedding map, and a somewhat cryptic remark that this map causes a splitting of the 2ndexterior power of the conormal module from Tor_1(I,R/I). In this talk we utilize differential graded algebra resolutions to define an explicit splitting map and show that if I/I^2 is free (or has enough free summands), then the 2nd exterior power of the conormal module (or its free summands) will split from Tor_1(I,R/I).

November 17th: Torkil Stai (NTNU), Perhaps Nakayama Algebra

Abstract: After semisimple ones, Nakayama algebras are the best understood algebras. But while their representation theory is under control, basic questions regarding their homological nature remain unanswered. We will look at both these features, insisting on accessibility for non-experts. Time permitting, we will also mention recent work with Fosse and Oppermann.

November 24th: No seminar-Thanksgiving Break