Algebraic Shifting Increases Relative Homology

Details Algebraic Shifting Increases Relative Homology Algebraic Shifting Increases Relative Homology

1998-09-30

arxiv.org/abs/math/9809195v1

Mathematics

Algebraic Topology

Scientific paper

\newcommand{\rhomi}[1]{\widetilde{H}_{#1}} \newcommand{\rbeti}[1]{\beta_{#1}} \newcommand{\kk}{\mathbf k} \newcommand{\dimk}{\dim_{\kk}} We show that algebraically shifting a pair of simplicial complexes weakly increases their relative homology Betti numbers in every dimension. More precisely, let $\Delta(K)$ denote the algebraically shifted complex of simplicial complex $K$, and let $\rbeti{j}(K,L)=\dimk \rhomi{j}(K,L;\kk)$ be the dimension of the $j$th reduced relative homology group over a field $\kk$ of a pair of simplicial complexes $L \subseteq K$. Then $\rbeti{j}(K,L) \leq \rbeti{j}(\Delta(K),\Delta(L))$ for all $j$. The theorem is motivated by somewhat similar results about Gr\"obner bases and generic initial ideals. Parts of the proof use Gr\"obner basis techniques.

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