System Supports for Teacher Learning and School Improvement

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 βj(K,L)=dimk H̃j(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 βj(K,L)⩽βj(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.