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# Categorical methods for algebra

**Alternative title:** Kategoriske metoder for algebra

#### Awarded: NOK 3.8 mill.

Source:

Project Manager:

Project Number:

302223

Application Type:

Project Period:

2020 - 2024

Funding received from:

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The scientific aims of the project have largely been met. Let us expand upon this claim bearing the initial popularized presentation in mind:
Generalized/partial Serre duality:
We introduced the concept of "partial Serre duality" for arbitrary triangulated categories, extending the classical notion of Serre duality in the Hom-finite case. One fundamental result, which prompted a surprisingly technical proof, is that partial Serre functors are triangulated. Another important result, in particular for the purposes of our project, is that for any Serre functor the domain consists of compact objects while the essential image consists --- up to direct summands --- of 0-cocompact objects. Adding applicability to the theory are also explicit constructions of generalized Serre functors for various flavors of homotopy categories. In particular the latter generalize well-known phenomena from Gorenstein homological algebra.
Representability results for functors:
When it comes to representability results for functors, we proved that dual Brown representability holds for any 0-cocompactly cogenerated triangulated category which sits at the base of a stable derivator. This last assumption is of course one which one could hope to get rid of, but on the other hand this has a silver lining: Our proof is constructive in the sense that we explicitly construct the representing objects.
Auslander--Reiten theory:
We have obtained generalizations of celebrated existence theorems for almost split triangles. In particular, results of Beligiannis and Krause which require the use of "local" duality have been subsumed by one which allows the use of a "global" duality (essentially a generalized Serre duality).
There are now several avenues which will be pursued in future research. In particular, it will be interesting to explore further the connection between various notions of (co)compactness and (co)-tilting.

Substantial scientific progress has been made, and more is on the way. It is too early to say much about the impact of the results in general.

The project is a mobility stipend for the project manager, who is to spend two years in Syracuse, NY, and a final year at NTNU Trondheim.
The most critical challenge is finding suitable research partners. The project team consists of leading experts in category theory, representation theory, and non-commutative geometry: The project manager, Steffen Oppermann (NTNU), Dan Zacharia (Syracuse), and Chrysostomos Psaroudakis (Thessaloniki). The members of the team are highly motivated to embark upon the research, and are chosen carefully, based on the extent to which their expertise is relevant to the primary and secondary project objectives, and ties to the project manager.
The objectives of the project are highly ambitious, and the successful execution of the project will fill obvious voids in contemporary mathematics.

1.6BILL. NOKtotal funding in the programme period752PROJECTShave received funding in the programme period1SOURCEhas financed the programme