141 research outputs found
A Lower Bound For Depths of Powers of Edge Ideals
Let be a graph and let be the edge ideal of . Our main results in
this article provide lower bounds for the depth of the first three powers of
in terms of the diameter of . More precisely, we show that \depth R/I^t
\geq \left\lceil{\frac{d-4t+5}{3}} \right\rceil +p-1, where is the
diameter of , is the number of connected components of and . For general powers of edge ideals we showComment: 21 pages, to appear in Journal of Algebraic Combinatoric
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
Cohen-Macaulay admissible clutters
There is a one-to-one correspondence between square-free monomial ideals and
clutters, which are also known as simple hypergraphs. It was conjectured that
unmixed admissible clutters are Cohen-Macaulay. We prove the conjecture for
uniform admissible clutters of heights 2 and 3. For admissible clutters of
greater heights, we give a family of examples to show that the conjecture may
fail. When the height is 4, we give an additional condition under which unmixed
admissible clutters are Cohen-Macaulay.Comment: 13 pages, final version to appear in J. Comm. Al
- …