# The Q-shaped derived category of a ring

Henrik Holm (University of Copenhagen)
Mathematics Seminar
Wednesday, 1 December, 2021 | 14:15–15:00 | G3.1 (1532-314)
Contact: Peter Jørgensen

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 special (but quite natural) small preadditive category $Q$ to Mod$(A)$. Note that 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.

Organised by: AarHomAlg