### 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.