The classical Dehn–Sommerville relations assert that the h‐vector of an Eulerian simplicial complex is symmetric. We establish three generalizations of the Dehn–Sommerville relations: one for the h‐vectors of pure simplicial complexes, another one for the flag h‐vectors of balanced simplicial complexes and graded posets, and yet another one for the toric h‐vectors of graded posets with restricted singularities. In all of these cases, we express any failure of symmetry in terms of “errors coming from the links.” For simplicial complexes, this further extends Klee's semi‐Eulerian relations.
A simplicial complex of dimension d − 1 is said to be balanced if its graph is d-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak Lefschetz property. As a corollary, we show that if ∆ is an arbitrary balanced triangulation of any closed homology manifold of dimension d − 1 ≥ 3, then 2h 2 (∆) − (d − 1)h 1 (∆) ≥ 4 d 2 (β 1 (∆) −β 0 (∆)), thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag h ′′ -vectors. d 1 +β 1 (∆).Equivalently, 2h 2 (∆) − (d − 1)h 1 (∆) ≥ 4 d 2 (β 1 (∆) −β 0 (∆)). Furthermore, if d ≥ 5, then this inequality holds as equality if and only if each connected component of ∆ is in the balanced Walkup class.
We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley-Reisner ring, demonstrating that the topological structure of the free action extends to the algebraic setting. If the complex in question is also Buchsbaum, this new description allows for a specialization of Schenzel's calculation of the Hilbert series of some of the ring's Artinian reductions. In further application, we generalize to the Buchsbaum case the results of Stanley and Adin that provide a lower bound on the h-vector of a Cohen-Macaulay complex admitting a free action by a cyclic group of prime order.
Given an infinite field and a simplicial complex ∆, a common theme in studying the f -and hvectors of ∆ has been the consideration of the Hilbert series of the Stanley-Reisner ring [∆] modulo a generic linear system of parameters Θ. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of h a d−1 (∆), the dimension over in degree d − 1 of [∆]/(Θ), for any complex ∆ of dimension d − 1. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.
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