Algebraic Shifting Increases Relative Homology

\newcommand{\rhomi}[1]{\widetilde{H}_{#1}} \newcommand{\rbeti}[1]{β_{#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 $Δ(K)$ 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 $\rbeti{j}(K,L) \leq \rbeti{j}(Δ(K),Δ(L))$ for all $j$. The theorem is motivated by somewhat similar results about Gröbner bases and generic initial ideals. Parts of the proof use Gröbner basis techniques.