Algebraic shifting increases relative homology (Q1591142)

Considering the algebraically shifted complex -- a remarkable procedure that finds for any simplicial complex \(K\) a shifted simplicial complex \(\Delta(K)\) combinatorially simpler with many of the same properties as \(K\) -- the author proves that this procedure always weakly increases the relative homology in every dimension. If \(\beta_j (K,L)= \dim_K\widetilde H_j (K,L;k)\) is the dimension of the \(j\)-th reduced relative homology group over a field \(k\) 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 proof of the main result uses Gröbner bases and generic initial ideals.