Depth of Powers of Squarefree Monomial Ideals (Research)

Lower bounds for the depth of second power of edge ideals

“…As it was mentioned in introduction, Fouli, Hà and Morey [6,7] determined a lower bound for the depth of I(G) in terms of the star packing number of G.…”

“…A star packing of G is a family X of stars in G which are pairwise disjoint, i.e., V (St(x)) ∩ V (St(x ′ )) = ∅, for St(x), St(x ′ ) ∈ X with x = x ′ . The quantity max |X | | X is a star packing of G is called the star packing number of G. Following [7], we denote the star packing number of G by α 2 (G).…”

“…Let T be the polynomial ring over K with variables corresponding to the vertices of H. It follows from [10, Corollary 1.6.3] that (7) depth…”

“…In [6], Fouli, Hà and Morey introduced the notion of initially regular sequence and used it to provide a method for estimating the depth of a homogenous ideal. Using this method, in [7], the same authors determined a combinatorial lower bound for the depth of edge ideals of graphs. Indeed, they proved that for every graph G with edge ideal I(G), we have depth S I(G) ≥ α 2 (G) + 1, where α 2 (G) denotes the so-called star packing number of G (see Section 2 for the definition of star packing number).…”

Regular Sequences On Squares Of Monomial Ideals

“…In the study of the depth function, it is desirable to know the depth of powers of an ideal instead of just the depth of the ideal itself (cf. [1,5,7,9,10,11,13,16]). This motivates the following natural question: when does a linear form or a sequence of linear forms constructed in [6] remain regular or initially regular with respect to powers of I?…”