A Lower Bound For Depths of Powers of Edge Ideals

Let $G$ be a graph and let $I$ be the edge ideal of $G$. Our main results in this article provide lower bounds for the depth of the first three powers of $I$ in terms of the diameter of $G$. More precisely, we show that \depth R/I^t \geq \left\lceil{\frac{d-4t+5}{3}} \right\rceil +p-1, where $d$ is the diameter of $G$, $p$ is the number of connected components of $G$ and $1 \leq t \leq 3$. For general powers of edge ideals we showComment: 21 pages, to appear in Journal of Algebraic Combinatoric

An example of an infinite set of associated primes of a local cohomology module

Let $(R,m)$ be a local Noetherian ring, let $I\subset R$ be any ideal and let $M$ be a finitely generated $R$-module. In 1990 Craig Huneke conjectured that the local cohomology modules $H^i_I(M)$ have finitely many associated primes for all $i$. In this paper I settle this conjecture by constructing a local cohomology module of a local $k$-algebra with an infinite set of associated primes, and I do this for any field $k$

Embedded Associated Primes of Powers of Square-free Monomial Ideals

An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all natural numbers t. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than beta_1, where beta_1 is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If in addition I fails to have the packing property, then embedded primes of I^t do occur when t=beta_1 +1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornu\'ejols.Comment: 15 pages, changes have been made to the title, introduction, and background material, and an example has been added. To appear in JPA