Algebraic Shifting Increases Relative Homology
Art M. Duval
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 $\beta_{j}(K,L)=\dim_{\boldsymbol{k}} \widetilde{H}_{j}(K,L;{\boldsymbol{k}})$ be the dimension of the $j\,$th reduced relative homology group over a field $\boldsymbol k$ 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 Gröbner bases and generic initial ideals. Parts of the proof use Gröbner basis techniques.