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

Abstract

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 nonzero 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.

Edge ideals with almost maximal finite index and their powers

“…where ∂ ∆(G) denotes the differentials of the chain complex of ∆(G). It is shown in [8,Theorem 3.2] that H 1 (∆(G); K) = 0 if and only if there exists a minimal cycle…”

“…In the second half of the paper we study the index of powers of edge ideals with almost maximal finite index. Although, for arbitrary ideals, many properties such as depth, projective dimension or regularity stabilize for large powers (see e.g [1,5,6,8,9,18,20,21,22]), their initial behaviour is often quite mysterious. However, edge ideals behave more controllable from the beginning.…”

On Generalization of Cycles and Chordality to Clutters from an Algebraic Viewpoint

“…By the Reisner theorem (see [8,Theorem 8.1.6]), ∆ is Cohen-Macaulay if and only if for all F ∈ ∆ and i < d F , H i (link ∆ F ; Z 2 ) = 0. But by [3,Proposition 5.1], for an x = F ∈C F ∈ C i (∆) we have x ∈ ker ∂ i if and only if C is a disjoint union of CF-cycles. Also it is easy to check that ∂ i (x) = F ∈∂(C) F .…”

“…But if we replace CF-cycle with C i -cycle, then we have: Proposition 4.7. Assume that i ∈ [3] and no subclutter of C is a C i -cycle, then C has a free MS. In particular, C is chordal and I(C) has a linear resolution.…”

“…(i) C = C ∪ {135, 235, 435, 635} is chordal and hence I(C ) has a linear resolution over every field. But C = C − {35} is an MS-induced non-complete C i -cycle (i ∈[3]) and also CF-cycle of C . (ii) If C = C ∪ {435}, then every non-empty vertex induced subclutter of C has a free MS.…”