On the derived category of finite posets
A finite poset can be seen as a finite category where the objects are the elements of the poset and the morphisms are given by the partial order relation. One can look at the category of functors from the poset to the category of vector spaces over a field. It is called the category of representations of the poset. By general arguments, it is equivalent to the category of modules over the incidence algebra of the poset.
The category of representation of a finite poset is abelian, so one can look at its bounded derived category. Since the incidence algebra has finite global dimension, this derived category has a Serre functor.
From the combinatorial point of view, this Serre functor is very important as it categorifies the usual Coxeter matrix of the poset.
In this talk, I will give various conjectures about the derived category of some famous posets and I will explain how one can construct some derived equivalences between derived categories of posets.
References
- On Derived Equivalences of Categories of Sheaves Over Finite Posets (ArXiv) by Ladkani.
- On the categories of modules over the Tamari posets (here) by Chapoton.
- On derived equivalences for categories of generalized intervals of a finite poset (ArXiv) by Chapoton, Ladkani and myself.
- Exceptional and modern intervals of the Tamari lattice (ArXiv) by myself.