CARS
This semester CARS was organized by Abbey Moos, Uyen Tran, and Stephen Stern
Dorian Kalir
Koszul Algebras and Koszul Duality
Sep 16
Abstract: Koszul duality was originally defined for graded algebras over a field and has seen many different generalizations. This is in part due to the many different ways Koszulness can be understood. In this talk we will examine a perspective guided by the twisted tensor construction which points to a generalization of Koszulness by Briggs, Cameron, Letz, and Pollitz from the field setting to morphisms of local rings.
Isaiah Martinez
The singularity category of a Gorenstein ring
Sep 9
Abstract: Given a Noetherian ring R, Ragnar-Olaf Buchweitz introduced the notion of the singularity category of R (under a different name). Buchweitz showed that when R is Gorenstein, the singularity category can be better understood through maximal Cohen-Macaulay modules and complete resolutions. We will briefly discuss the history of Buchweitz's manuscript, present the main result that brings the singularity category of a Gorenstein ring "down to earth," and (if time permits) discuss the significance and impact of Buchweitz's manuscript.
Zach Nason
Quasi-Gorenstein Morphisms II: Electric Boogaloo
Sep 2
Abstract: On Sunday, I talked about joint work with Andrew Soto Levins and Ryan Watson that characterized quasi-Gorenstein morphisms of DG-rings using the virtually G-small property. In CARS today, I'm going to go over the other half of our work, which deals with exact sequences in noetherian local rings, and how to detect exact sequences by measuring whether certain maps of Koszul complexes are quasi-Gorenstein.
Kesavan Mohana Sundaram
On the annihilators of local cohomology modules
Aug 26
Abstract: In 2011, Lynch conjectured that the Krull dimension of the top local cohomology module supported at an ideal I in a Noetherian local ring R, is same as the dimension of R/T, where T is the I-torsion submodule of R. In 2019, Hochster raised a question that, is the top local cohomology module of a local Noetherian domain with support in a given ideal faithful? In the recent years, some cool progress and counterexamples have been made on these questions. So, why not check them out?
