### Abstract

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 $\beta_{j}(K,L)=\dimk
\rhom_{j}(K,L;\kk)$ 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
$\beta_{j}(K,L) \leq \beta_{j}(\Delta(K),\Delta(L))$ for all $j$.

The theorem is motivated by somewhat similar results about Grobner
bases and generic initial ideals. Parts of the proof use Grobner
basis techniques.