On homological dimensions and combinatorics of finite dimensional algebras

• Applicant: Dr. Rene Marczinzik
• Subject Area: Mathematics
• Term: from 2019 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 428999796

• Final Report Year: 2022

Final Report Abstract

In this project we studied connections between homological algebra of finite dimensional algebras and combinatorial objects. The most important result is a new connection between order theory and homological algebra that was found in joint work with Osamu Iyama. We give a homological characterisation when a finite lattice L is distributive and use this homological characterisation to show that the global dimension, the most important homological dimension, of the incidence algebra of L coincides with the order dimension of L, which is the most important order theoretic dimension. We use our results to give a categorification of the rowmotion map, which is the main attraction in the recent field of dynamical algebraic combinatorics. In joint work with Thomas and Yildirim we study the interaction of this rowmotion bijection with the Coxeter transformation of the incidence algebra of a distributive lattice. We show that they satisfy a mysterious identity that is also present in other situations involving higher representation-finite algebras. In joint work with Chan, Darpö and Iyama we give a new connection to the class of fractionally Calabi-Yau algebras, a categorical notion that appears in many areas in algebra, and certain periodic Frobenius algebra. The new methods that we introduce are strong enough to answer several open problems such as the construction of wild periodic algebras of arbitrary high period and the derived invariance of twisted fractionally Calabi-Yau algebras. We also profited from the use of computer experiments with the GAP-package QPA that helped us to o answer two open questions of Erdmann and Holm. In joint work with Böhmler, we showed the existence of a cluster-tilting object in a representation-infinite block of a group algebra, which is the first example of a cluster tilting module for this class of algebras. In joint work with Vaso we found an example of an algebra with a 2-cluster tilting object which does not have finite complexity, which shows that the result of Erdmann and Holm on selfinjective algebras can not be extended to the class of all finite dimensional algebras. In joint work with Cruz, we show that a properly stratified algebra is Gorenstein if and only if the characteristic tilting module coincides with the characteristic cotilting module and use this result to describe Ringel duals for special classes of algebras including for example centraliser algebras of nilpotent matrices. In joint work with Madsen and Zaimi, we construct Nakayama algebras that are new classes of higher Auslander algebras and show how they can be used to obtain optimal bounds for inequalites involving the global dimension of Nakayama algebras. We also found a counterexample to a 20 year old conjecture of Reineke on stability conditions for Dynkin quivers that was found with the help of the computer.

Publications

• On the classification of higher Auslander algebras for Nakayama algebras. Journal of Algebra, 556, 776-805.
  Madsen, Dag Oskar; Marczinzik, René & Zaimi, Gjergji
  
• On properly stratified Gorenstein algebras. Journal of Pure and Applied Algebra, 225(12), 106757.
  Cruz, Tiago & Marczinzik, René
  
• A cluster tilting module for a representation-infinite block of a group algebra. Journal of Algebra, 589, 483-494.
  Böhmler, Bernhard & Marczinzik, René
  
• Distributive lattices and Auslander regular algebras. Advances in Mathematics, 398, 108233.
  Iyama, Osamu & Marczinzik, René
  
• Existence of a 2-cluster tilting module does not imply finite complexity. Journal of Algebra, 598, 385-391.
  Marczinzik, René & Vaso, Laertis
  
• On total stability conditions for Dynkin quivers. Bulletin of the London Mathematical Society, 55(2), 886-891.
  Marczinzik, René
  
• On the interaction of the Coxeter transformation and the rowmotion bijection. Journal of Combinatorial Algebra, 8(3), 359-374.
  Marczinzik, René; Thomas, Hugh & Yıldırım, Emine
  
• Periodic trivial extension algebras and fractionally Calabi-Yau algebras. Annales Scientifiques de l'École Normale Supérieure.
  CHAN, Aaron; DARPÖ, Erik; IYAMA, Osamu & MARCZINZIK, René