The Q-shaped derived category of a ring

Abstract: The derived category D(A) of the category Mod(A) of modules over a ring A is an important example of a triangulated category in algebra. It can be obtained as the homotopy category of the category Ch(A) of chain complexes of A-modules equipped with its standard model structure. One can view Ch(A) as the category Fun(Q,Mod(A)) of additive functors from a certain small preadditive category Q to Mod(A). The model structure on Ch(A) = Fun(Q,Mod(A)) is not inherited from a model structure on Mod(A) but arises instead from the "self-injectivity" of the special category Q. We will show that the functor category Fun(Q,Mod(A)) has two interesting model structures for many other self-injective small preadditive categories Q. These two model structures have the same weak equivalences, and the associated homotopy category is what we call the Q-shaped derived category of A. We will also show that it is possible to generalize the homology functors on Ch(A) to homology functors on Fun(Q,Mod(A)) for most self-injective small preadditive categories Q. The talk is based on a joint paper with Peter Jørgensen (arXiv:2101.06176), which has the same title as the talk.

rings and algebrasrepresentation theory

Audience: researchers in the topic

( paper )

ONCAS Online Noncommutative Algebra Seminar

Series comments: This is an online seminar organized to keep the community of noncommutative algebraists connected during the pandemic which has made it difficult for everyone to travel. If you would like to attend these talks, please email to any of the organizers and you will be added to the mailing list.