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We consider a sheaf of exterior algebras on a simplicial poset $S$ andintroduce a notion of homological characteristic function. Two natural objectsare associated with these data: a graded sheaf $\mathcal{I}$ and a gradedcosheaf $\widehat{\Pi}$. When $S$ is a homology manifold, we prove theisomorphism $H^{n-1-p}(S;\mathcal{I})\cong H_{p}(S;\widehat{\Pi})$ which can beconsidered as an extension of the Poincare duality. In general, there is aspectral sequence $E^2_{p,q}\cong H^{n-1-p}(S;\mathcal{U}_{n-1+q}\otimes\mathcal{I})\Rightarrow H_{p+q}(S;\widehat{\Pi})$, where $\mathcal{U}_*$ is thelocal homology stack on $S$. This spectral sequence, in turn, extendsZeeman--McCrory spectral sequence. This sheaf-theoretical result is applied totoric topology. We consider a manifold $X$ with a locally standard action of acompact torus and acyclic proper faces of the orbit space. A principal torusbundle $Y$ is associated with $X$, so that $X\cong Y/\sim$. The orbit typefiltration on $X$ is covered by the topological filtration on $Y$. We provethat homological spectral sequences associated with these two filtrations areisomorphic in many nontrivial positions.

Locally standard torus actions and sheaves over Buchsbaum posets