arXiv subjects

Emma Connon

Publications and source records attributed to Emma Connon.

3 recordsLinked to original sources

arXiv2013

On $d$-dimensional cycles and the vanishing of simplicial homology

In this paper we introduce the notion of a $d$-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use these results to describe the relationship between the combinatorial structure of a simplicial complex and its simplicial homology. In particular, we show that over any field of characteristic 2 the existence of non-zero $d$-dimensional homology corresponds exactly to the presence of a $d$-dimensional cycle in the simplicial complex. We also show that $d$-dimensional cycles which are orientable give rise to non-zero simplicical homology over any field.

math.AT↗

arXiv2013

Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution

In this paper we extend one direction of Fröberg's theorem on a combinatorial classification of quadratic monomial ideals with linear resolutions. We do this by generalizing the notion of a chordal graph to higher dimensions with the introduction of d-chorded and orientably-d-cycle-complete simplicial complexes. We show that a certain class of simplicial complexes, the d-dimensional trees, correspond to ideals having linear resolutions over fields of characteristic 2 and also give a necessary combinatorial condition for a monomial ideal to be componentwise linear over all fields.

math.AC↗

arXiv2013

A criterion for a monomial ideal to have a linear resolution in characteristic 2

In this paper we give a necessary and sufficient combinatorial condition for a monomial ideal to have a linear resolution over fields of characteristic 2. We also give a new proof of Fröberg's theorem over fields of characteristic 2.

math.AC↗