Higher homological methods in karoubian categories

• Applicant: Privatdozent Dr. Sven-Ake Wegner
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 507660524

Homological algebra has a short but outstanding history. Invented in the 1960s, by today its methods help in many areas of mathematics to unveil the common principle of similar phenomena that appear in different contexts. Modern presentations often start with an abelian category and introduce its derived category. Techniques like derived functors and classical homological invariants can then be defined in large generality. Derived equivalences, i.e., the equivalence of the derived categories, can additionally be interpreted as a higher form of invariance. A powerful tool to establish derived equivalences between abelian categories is given by tilting theory. The latter was invented in the 1980s in the context of representations of finite dimensional algebras, but was very quickly identified as a general tool in homological algebra.The motivation of the current project comes from functional analysis. Here, homological methods emerged during the 1970s in the form of derived functors. Derived categories appeared around 2000 but have been used since then only occasionally. A major reason for this is the fact that in functional analysis the ambient categories are never abelian. A comprehensive theory of the derived category is available only for so-called quasiabelian categories which cover for example Banach spaces, Fréchet spaces or bornological spaces. Treating further classical objects of functional analysis (real analytic functions, distributions, test functions but also bornological modules) requires a more general category theory.In this project we will focus on the class of so-called karoubian categories. Here the derived category is known to be well-defined and there is evidence that many functional analytic scenarios give rise to categories of this type. Our first aim is to gain new insights into the derived category of a karoubian category and make it accessible to computations. Secondly we will develop methods to construct derived equivalences and we will illustrate applications via the categorification of functional analytic problems. Thirdly, we will develop a tilting theory for categories of bornological modules. The latter actually are quasiabelian but as it is more natural to consider a non-maximal exact structure on them, they fit better in a theory of karoubian categories as suggested above. In this branch of the project we first aim to prove an analogue of the classic Brenner-Butler theorem. Then we will investigate how derived equivalences can be induced by tilting modules.

• DFG Programme: Research Grants