24 research outputs found
Replication in critical graphs and the persistence of monomial ideals
Motivated by questions about square-free monomial ideals in polynomial rings,
in 2010 Francisco et al. conjectured that for every positive integer k and
every k-critical (i.e., critically k-chromatic) graph, there is a set of
vertices whose replication produces a (k+1)-critical graph. (The replication of
a set W of vertices of a graph is the operation that adds a copy of each vertex
w in W, one at a time, and connects it to w and all its neighbours.)
We disprove the conjecture by providing an infinite family of
counterexamples. Furthermore, the smallest member of the family answers a
question of Herzog and Hibi concerning the depth functions of square-free
monomial ideals in polynomial rings, and a related question on the persistence
property of such ideals
Generalized cover ideals and the persistence property
Let be a square-free monomial ideal in , and
consider the sets of associated primes for all integers . Although it is known that the sets of associated primes of powers of
eventually stabilize, there are few results about the power at which this
stabilization occurs (known as the index of stability). We introduce a family
of square-free monomial ideals that can be associated to a finite simple graph
that generalizes the cover ideal construction. When is a tree, we
explicitly determine for all . As consequences, not
only can we compute the index of stability, we can also show that this family
of ideals has the persistence property.Comment: 15 pages; revised version has a new introduction; references updated;
to appear in J. Pure. Appl. Algebr
The normalized depth function of squarefree powers
The depth of squarefree powers of a squarefree monomial ideal is introduced.
Let be a squarefree monomial ideal of the polynomial ring
. The -th squarefree power of is the
ideal of generated by those squarefree monomials with each
, where is the unique minimal system of monomial generators
of . Let denote the minimum degree of monomials belonging to
. One has . Setting
, one calls the
normalized depth function of . The computational experience strongly invites
us to propose the conjecture that the normalized depth function is
nonincreasing. In the present paper, especially the normalized depth function
of the edge ideal of a finite simple graph is deeply studied
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
The uniform face ideals of a simplicial complex
We define the uniform face ideal of a simplicial complex with respect to an
ordered proper vertex colouring of the complex. This ideal is a monomial ideal
which is generally not squarefree. We show that such a monomial ideal has a
linear resolution, as do all of its powers, if and only if the colouring
satisfies a certain nesting property.
In the case when the colouring is nested, we give a minimal cellular
resolution supported on a cubical complex. From this, we give the graded Betti
numbers in terms of the face-vector of the underlying simplicial complex.
Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both
the ideal and its quotient. We also give explicit formul\ae\ for the
codimension, Krull dimension, multiplicity, projective dimension, depth, and
regularity. Further still, we describe the associated primes, and we show that
they are persistent.Comment: 34 pages, 8 figure
In the shadows of a hypergraph: looking for associated primes of powers of square-free monomial ideals
The aim of this paper is to study the associated primes of powers of square-free monomial ideals. Each square-free monomial ideal corresponds uniquely to a finite simple hypergraph via the cover ideal construction, and vice versa. Let H be a finite simple hypergraph and J(H) the cover ideal of H. We define the shadows of hypergraph, H, described as a collection of smaller hypergraphs related to H under some conditions. We then investigate how the shadows of H preserve information about the associated primes of the powers of J(H). Finally, we apply our findings on shadows to study the persistence property of square-free monomial ideals and construct some examples exhibiting failure of containment