Dualizing Complex of the Incidence Algebra of a Finite Regular Cell Complex

Let Σ be a finite regular cell complex with ∅ ∈ Σ, and regard it as a poset (i.e., partially ordered set) by inclusion. Let R be the incidence algebra of the poset Σ over a field k. Corresponding to the Verdier duality for constructible sheaves on Σ, we have a dualizing complex ω ∈ Db(modR⊗kR) giving a duality functor from Db(modR) to itself. This duality is somewhat analogous to the Serre duality for projective schemes (∅ ∈ Σ plays a similar role to that of “irrelevant ideals”). If H(ω) 6= 0 for exactly one i, then the underlying topological space of Σ is Cohen-Macaulay (in the sense of the Stanley-Reisner ring theory). The converse also holds if Σ is a meet-semilattice as a poset (e.g., Σ is a simplicial complex). R is always a Koszul ring with R ∼= R. The relation between the Koszul duality for R and the Verdier duality is discussed. This result is a variant of a theorem of Vybornov.