Dualizing complex of the incidence algebra of a finite regular cell complex

Abstract

Let $\Sigma$ be a finite regular cell complex with $\emptyset \in \Sigma$, and regard it as a partially ordered set (poset) by inclusion. Let $R$ be the incidence algebra of the poset $\Sigma$ over a field $k$. Corresponding to the Verdier duality for constructible sheaves on $\Sigma$, we have a dualizing complex $w \in D^b(mod_{R \otimes_k R})$ giving a duality functor from $D^b(mod_R)$ to itself. $w$ satisfies the Auslander condition. Our duality is somewhat analogous to the Serre duality for projective schemes ($\emptyset$ plays a similar role to that of "irrelevant ideals"). If $H^i(w) \ne 0$ for exactly one $i$, then the underlying topological space of $\Sigma$ is Cohen-Macaulay (in the sense of the Stanley-Reisner ring theory). The converse also holds when $\Sigma$ is a simplicial complex. $R$ is always a Koszul ring with $R^! \cong R^op$. The relation between the Koszul duality for $R$ and the Verdier duality is discussed. This result is a variant of a theorem of Vybornov. The Mobius function of the poset $\hat{\Sigma}$ is also discussed.