有限正则胞腔复形的关联代数的对偶复形
Dualizing complex of the incidence algebra of a finite regular cell complex
摘要：
设$Σ$是一个含有$\emptyset \in Σ$的有限正则胞腔复形，通过包含将其看作偏序集.设$R$是偏序集$Σ$在域$k$上的关联代数.对应于$Σ$上可构造层的Verdier对偶，我们有一个对偶复形$w \in D^b(mod_{R \otimes_k R})$，给出从$D^b(mod_R)$到它自身的对偶函子。$w$满足Auslander条件。我们的对偶有点类似于投射方案的Serre对偶（$\emptyset$起着类似于“无关理想”的作用）。如果$H^i(w) \ne 0$恰好对一个$i$，则$Σ$的底层拓扑空间是Cohen-Macaulay（在Stanley-Reisner环理论的意义下）。当$Σ$是单纯复形时，反之也成立。$R$总是一个具有$R^! \cong R^op$的Koszul环。讨论了$R$的Koszul对偶与Verdier对偶之间的关系。这一结果是一个变种定理的维博尔诺夫。讨论了偏序集$\hatΣ$的Mobius函数.
英文摘要：
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.
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