Stanley-Reisner rings, sheaves, and Poincare-Verdier duality

Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomial ideals. The notion of Alexander duality is generalized here to arbitrary monomial ideals. It is shown how this duality is naturally expressed by Bass numbers, in their relations to the Betti numbers of a monomial ideal and its Alexander dual. Relative cohomological constructions on...

We consider a sheaf of exterior algebras on a simplicial poset $S$ and introduce a notion of homological characteristic function. Two natural objects are associated with these data: a graded sheaf $\mathcal{I}$ and a graded cosheaf $\widehat{\Pi}$. When $S$ is a homology manifold, we prove the isomorphism $H^{n-1-p}(S;\mathcal{I})\cong H_{p}(S;\widehat{\Pi})$ which can be considered as an extension of the Poincare duality. In general, there is a spectral sequence $E^2_{p,q}\c...

Let $\Delta$ be simplicial complex and let $k[\Delta]$ denote the Stanley--Reisner ring corresponding to $\Delta$. Suppose that $k[\Delta]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert--Samuel multiplicity of $k[\Delta]$ in terms of the $h$--vector of $\Delta$. As an application, we derive a linear equation system and some inequalities for the components of the $h$--vector of the clique complex of an arbitrary chordal graph. As an other appli...

Let $X$ be a finite connected simplicial complex, and let $\delta$ be a perversity (i.e., some function from integers to integers). One can consider two categories: (1) the category of perverse sheaves cohomologically constructible with respect to the triangulation, and (2) the category of sheaves constant along the perverse simplices ($\delta$-sheaves). We interpret the categories (1) and (2) as categories of modules over certain quadratic (and even Koszul) algebras $A(X,\de...

In this paper, we prove that the Stanley--Reisner ideal of any connected simplicial complex of dimension $\ge 2$ that is locally complete intersection is a complete intersection ideal. As an application, we show that the Stanley--Reisner ideal whose powers are Buchsbaum is a complete intersection ideal.

We define a ring R of geometric objects G generated by finite abstract simplicial complexes. To every G belongs Hodge Laplacian H as the square of the Dirac operator determining its cohomology and a unimodular connection matrix L). The sum of the matrix entries of the inverse of L is the Euler characteristic. The spectra of H as well as inductive dimension add under multiplication while the spectra of L multiply. The nullity of the Hodge of H are the Betti numbers which can n...

Let $\Delta$ be a simplicial complex. We study the expansions of $\Delta$ mainly to see how the algebraic and combinatorial properties of $\Delta$ and its expansions are related to each other. It is shown that $\Delta$ is Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum or $k$-decomposable, if and only if an arbitrary expansion of $\Delta$ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley-Reisner...

An alternative proof of bornological Verdier duality for complex manifolds, as proven initially by Prosmans & Schneiders is given, using Schneider's theory of quasi-abelian homological algebra, and the theory of residues and duality.

We give a geometric interpretation of the Stanley--Reisner correspondence, extend it to schemes, and interpret it in terms of the field of one element.

A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley-Reisner ring $R/I_\Lambda$ and the inverse system algebra $R/I_\Delta$. We give a careful study of both of these algebras. Our main results are a full description of the graded Betti numbers of both algebras in the more general setting of linear spaces (giving the result for the projective planes as a special case), and a...