Algebraic shifting increases relative homology

We show that algebraically shifting a pair of simplicial complexes weakly increases their relative homology Betti numbers in every dimension. More precisely, let ∆(K) denote the algebraically shifted complex of simplicial complex K, and let β j (K, L) = dim k H j (K, L; k) be the dimension of the jth reduced relative homology group over a field k of a pair of simplicial complexes L ⊆ K. Then β j (K, L) ≤ β j (∆(K), ∆(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.