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Sequentially Cohen-Macaulay edge ideals
Let G be a simple undirected graph on n vertices, and let I(G) C R = k[x 1 ,...,x n ] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof
Splittings of monomial ideals
We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and
Associated primes of monomial ideals and odd holes in graphs
Let G be a finite simple graph with edge ideal I(G). Let I(G)∨ denote the Alexander dual of I(G). We show that a description of all induced cycles of odd length in G is encoded in the associated
Generalizing the Borel property
The notion of Q-Borel ideals is introduced: ideals which are closed under the Borel moves arising from a poset Q, and evidence that they interpolate between Borel ideals and arbitrary monomial ideals is offered.
Lex-Plus-Powers Ideals
Hilbert showed in his 1890 paper [Hilbert] that one can compute the Hilbert function from a graded free resolution (or simply the set of graded Betti numbers). Thus there is an (easy) combinatorial
Asymptotic resurgence via integral closures
Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic