Dualizing complex of the incidence algebra of a finite regular cell complex
Abstract
Description
Let $Σ$ be a finite regular cell complex with $\emptyset \in Σ$, and regard it as a partially ordered set (poset) 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 $w \in D^b(mod_{R \otimes_k R})$ 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 $Σ$ is Cohen-Macaulay (in the sense of the Stanley-Reisner ring theory). The converse also holds when $Σ$ 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 $\hatΣ$ is also discussed.
18 pages. The results are almost same. But the exposition has been totally revised emphasizing combinatorial aspects
18 pages. The results are almost same. But the exposition has been totally revised emphasizing combinatorial aspects