Chain Complexes over Principal Ideal Domains

Abstract

Chain complexes are studied here as an abstract algebraic generalisation of geometric simplicial complexes. From this point of view, we extend the notions of shellability and of a cone, which are both defined for simplicial complexes, to chain complexes. We define a cone for chain complexes in an abstract way abandoning the geometrical idea of an apex and compare it with mapping cones. Indeed, there are cones which cannot be regarded as mapping cones, in contrast to the simplicial case. And conversely, we name certain conditions on which a mapping cone is a cone matching our definition. Our notion of shellability given here for chain complexes is a generalisation of this well-known term which is defined for simplicial complexes as well as for regular finite CW-complexes. But in contrast to shellable simplicial complexes, there is no information about the homology of shellable chain complexes, so we claim additional conditions on them which imitate other properties of simplicial complexes. This leads to our notions of regular and totally regular chain complexes. We obtain complete homological information for totally regular chain complexes which have a specific augmentation map. In the end, we consider mapping cones over shellable or regular chain complexes and show that they also are shellable or regular, respectively.