143 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
Hilbert Coefficients and Sally Modules: A Survey of Vasconcelos' Contributions
This paper surveys and summarizes Wolmer Vasconcelos' results surrounding
multiplicities, Hilbert coefficients, and their extensions. We particularly
focus on Vasconcelos' results regarding multiplicities and Chern coefficients,
and other invariants which they bound. The Sally module is an important
instrument introduced by Vasconcelos for this study, which naturally relates
Hilbert coefficients to reduction numbers.Comment: 30 page
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