On the index of powers of edge ideals

Abstract

The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal I is strictly increasing if I has linear relations. Examples show that this need not to be the case for monomial ideals generated in degree greater than two.

Squarefree Powers of Edge Ideals of Forests

“…If k > mat(G), then I(G) [k] = (0). The study of squarefree powers was initiated in [4] and continued in [5]. Our motivation to study such powers is twofold.…”

“…The following lemma shows the existence of a certain kind of leaf in forests. The authors of [4] proved the surprising result that the highest non-vanishing squarefree power of an edge ideal has linear resolution. This result will be crucial in the proof of Theorem 25.…”

Edge ideals with almost maximal finite index and their powers

“…For the edge ideals with linear resolution all non-linear Betti numbers are zero. For the edge ideals with maximal finite index t, it is seen in [11,2] that there is only one nonzero non-linear Betti number β t,t+3 (I) = 1 over all fields. In the case of ideals with almost maximal finite index with index(I) = t, the non-linear Betti numbers appear in the last two homological degrees of the minimal free resolution.…”

“…If the index of I attains the largest finite value, we have index(I) = pd(I), where pd(I) denotes the projective dimension of I. In this case the ideal I is said to have maximal finite index, see [2]. In [2,Theorem 4.1], it was shown that the edge ideal I(G) has maximal finite index if and only if Ḡ is an induced cycle of length > 3.…”

“…In this case the ideal I is said to have maximal finite index, see [2]. In [2,Theorem 4.1], it was shown that the edge ideal I(G) has maximal finite index if and only if Ḡ is an induced cycle of length > 3. In this paper, we proceed one more step and consider the edge ideals I(G) with index(I(G)) = pd(I(G)) − 1.…”