ALGEBRAIC SHIFTING INCREASES RELATIVE HOMOLOGY

ART M. DUVAL

Abstract. We show that algebraically shifting a pair of simplicial complexes weakly increases their relative homology Betti numbers in every dimension.

More precisely, let Δ(K) denote the algebraically shifted complex of simplicial complex K, and let β_{j}(K, L) = dim_{k} H̃_{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}(Δ(K), Δ(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.