Algebraic Shifting Increases Relative Homology
  \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. 
Art M. Duval