Algebraic h-vectors of simplicial complexes through local cohomology, part 1

2019/07/29 by Connor Sawaske, Sawaske, Connor
Mathematics · #05E45 #14F55 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.CO #msc:05E45 #msc:14F55

paper · pdf · doi:10.48550/arxiv.1907.12620

arxiv created 2019/07/29 · arxiv updated 2019/07/31

Abstract

Given an infinite field \mathbbk and a simplicial complex Δ, a common theme in studying the f- and h-vectors of Δ has been the consideration of the Hilbert series of the Stanley--Reisner ring \mathbbk[Δ] 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_d-1^\mathfraka(Δ), the dimension over \mathbbk in degree d-1 of \mathbbk[Δ]/(Θ), for any complex Δ 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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