Ladders of recollements of triangulated and of abelian categories

• Applicant: Professor Dr. Steffen Koenig
• Subject Area: Mathematics
• Term: from 2017 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 340487543

This proposal is about recollements of triangulated and of abelian categories. Triangulated recollements have been introduced by Beilinson, Bernstein and Deligne to deconstruct triangulated categories into open and closed parts, related for instance by Grothendieck's six functors; recollements can be seen as short exact sequences of three triangulated (such as derived) categories, hence also as a far reaching generalisation of the connections between two algebras provided by derived equivalences . Ladders of triangulated categories are recollements extended by further adjoints; these have been introduced by Beilinson, Ginzburg and Schechtman as a setup for Koszul duality. Recollements of abelian categories have been introduced, too, but ladders of abelian categories need to be defined rather differently; this is to be done in this project. The project then aims at connecting and combining triangulated and abelian techniques, at further developing the method of ladders and at a variety of intrinsically related applications.The intended applications include- to characterise, describe and construct algebras frequently arising in algebraic Lie theory (such as quasi-hereditary algebras) as well as important representations (such as characteristic tilting modules);- to relate ladders with Serre functors and to use them to construct such functors;- to fully describe recollements of self-injective (eg symmetric or Frobenius) algebras and to use this description to classify self-injective (or Frobenius or symmetric) cellularly stratified diagram algebras (such as Brauer, BMW or partition algebras);- to relate being Gorenstein and validity of the (Fg) condition for algebras in a recollement and to describe support varieties of modules or complexes under the functors in a recollement.

• DFG Programme: Research Grants