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تسجيل مستخدم جديدStanley-Reisner rings of Buchsbaum complexes with a free action by an abelian group
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Connor Sawaske
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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 Schenzels calculation of the Hilbert series of some of the rings 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.
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The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d - 1$. We prove that the inequality $d leq mathrm{reg}(Delta) cdot mathrm{type}(Delta)$ holds for a
ny $(d-1)$-dimensional Cohen-Macaulay simplicial complex $Delta$ satisfying $Delta=mathrm{core}(Delta)$, where $mathrm{reg}(Delta)$ (resp. $mathrm{type}(Delta)$) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring $Bbbk[Delta]$. Moreover, for any given integers $d,r,t$ satisfying $r,t geq 2$ and $r leq d leq rt$, we construct a Cohen-Macaulay simplicial complex $Delta(G)$ as an independent complex of a graph $G$ such that $dim(Delta(G))=d-1$, $mathrm{reg}(Delta(G))=r$ and $mathrm{type}(Delta(G))=t$.
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Let $Delta$ be a triangulated homology ball whose boundary complex is $partialDelta$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $Delta$, $mathbb F[Delta]$, is isomorphic to the Stanley--Reisner module of t
he pair $(Delta, partialDelta)$, $mathbb F[Delta,partial Delta]$. This result implies that an Artinian reduction of $mathbb F[Delta,partial Delta]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $mathbb F[Delta]$. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the $h$-numbers of Buchsbaum complexes and use it to prove the monotonicity of $h$-numbers for pairs of Buchsbaum complexes as well as the unimodality of $h$-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold $g$-conjecture.
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