Triangulated Categories in Representation Theory

We are organizing a one-day meeting with three talks on recent research developments pertaining to various triangulated categories arising in representation theory. I am giving one of the talks myself and the other two speakers are Sergio Estrada from the University of Murcia and Tsutomu Nakamura from the University of Tokyo. All the talks will be held at the following Zoom coordinates - 930 5372 8057 (there is no passcode).  

Please feel free to share the news about this meeting within your departments or with anyone who might be interested in attending. If you have any questions, comments or queries, please email me at rudradip.biswas@manchester.ac.uk or rudradipbiswas@gmail.com.  

Dr. Rudradip Biswas, Organizer. 

(Please refer to my website for details about me, my research and other seminars organized by me)

Post-conference note: The talks are now available on the YouTube channel of the conference. They have also been linked below. Thank you everyone for your participation.

We tried to take a (virtual) conference picture. However, since we only announced that we would take a picture after the end of the last talk, many attendees had exited the Zoom meeting by the point the picture was taken. So, please note that those who can be seen in the picture below weren't the only people in attendance. 

Date of the Meeting:  November 11, 2021; Thursday. 

Zoom ID: 930 5372 8057. https://zoom.us/j/93053728057

Time: 10 am GMT (11 am Berlin/Paris/Madrid)

Speaker: Tsutomu Nakamura (University of Tokyo, Japan)

Title:  Flat cotorsion modules over Noether algebras and derived categories.

Abstract: For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. We then explain several important roles of complexes of flat cotorsion modules and give some applications. The first half of this talk is based on joint work with Ryo Kanda and the second half is partly based on joint work with Peder Thompson (J. Algebra 562 (2020), 587--620).  

A video recording of the above talk is linked below: 

Time: 11 am GMT (12 noon Berlin/Paris/Madrid)

Speaker: Rudradip Biswas (Tata Institute of Fundamental Research - Mumbai, India)

Title: Injective generation of the derived category for group algebras.

Abstract: In a recent paper, Jeremy Rickard showed that for any finite-dimensional algebra R over a field, if the R-injectives generate the derived category D(R), then the finitistic dimension of R is finite (recall that the famous finitistic dimension conjecture claims that for any R that is a finite-dimensional algebra over a field, the finitistic dimension of R should be finite). In the same paper, it was noted that there are no known finite-dimensional algebras over fields for which this injective generation property is absent. In this talk, we will consider the case when R is a group algebra (not necessarily a finite-dimensional algebra over a field), and show that for a large class of groups, the finiteness of the finitistic dimension of the group algebra (over any commutative ring of finite global dimension) implies the above injective generation property. We will also show how this question, for group algebras, is very closely connected to some existing conjectures on various cohomological invariants for groups, and that will lead us to a version of the finitistic dimension conjecture for group algebras.

A video recording of the above talk is linked below:

Note that the papers mentioned as [Bi1], [Bi2], [Bi3] in my presentation, as seen in the video below, are respectively the articles numbered [1], [4], [2] in the "Papers and Preprints" section of my website. 

BREAK: 12 noon - 12.30 pm GMT (1 pm - 1.30 pm Berlin/Paris/Madrid)

Time: 12.30 pm GMT (1.30 pm Berlin/Paris/Madrid)

Speaker: Sergio Estrada (University of Murcia, Spain)

Title: The singularity category of an exact category - applications.  

Abstract: We consider (big) singularity categories and Gorenstein defect categories in the setting of exact categories, and especially in the presence of a complete hereditary cotorsion pair. As a main result, we show that the vanishing of this more general Gorenstein defect category characterizes finiteness of certain Gorenstein dimensions and provide an equivalence of a big singularity category with the stable category of Gorenstein objects. Applications include a viable non-affine analogue of the (big) singularity category of a ring and a perspective on the finitistic dimension conjecture.

A video recording of the above talk is linked below.

The papers that the speaker mentioned at the start of the presentation, as one can see in the video below, are the following (click on the numbers): [1], [2], [3], [4], [5].