Lower bounds are given for the depths of R/I^t for t at least one when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.
We survey recent studies and results on the following problem: which numerical functions can be the depth function of powers and symbolic powers of homogeneous ideals.
The aim of this paper is to give a formula for the Stanley depth of quotient of powers of the path ideal. As a consequence, we establish that the behaivior of the Stanley depth of quotient of powers of the path ideal is the same as a classical result of Brodmann on depth.
We compute the depth and Stanley depth for the quotient ring of the path ideal of length $3$ associated to a $n$-cyclic graph, given some precise formulas for depth when $n\not\equiv 1\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1\,(\mbox{mod}\ 4)$ and for Stanley depth when $n\equiv 0,3\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1,2\,(\mbox{mod}\ 4)$. Also, we give some formulas for depth and Stanley depth of a quotient of the path ideals of length $n-1$ and $n$.
In this paper we study depth and Stanley depth of the edge ideals and quotient rings of the edge ideals, associated to classes of graphs obtained by taking the strong product of two graphs. We consider the cases when either both graphs are arbitrary paths or one is an arbitrary path and the other is an arbitrary cycle. We give exact formulae for values of depth and Stanley depth for some sub classes. We also give some sharp upper bounds for depth and Stanley depth in the general cases.
Let $S$ be a ring of polynomials in finitely many variables over a field. In this paper we give lower bounds for depth and Stanley depth of modules of the type $S/I^t$ for $t\geq1$, where $I$ is the edge ideal of some caterpillar and lobster trees. These new bounds are much sharper than the existing bounds for the classes of ideals we considered.
We consider the face ideal associated to a line-type simplicial complex. We compute the \texttt{depth} and the \texttt{sdepth} for its quotient ring. In particular, the facet ideal and its quotient ring satisfy the Stanley inequality.
Let $k$ be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the $k^{th}$ power of a path on $n$ vertices. We show that both depth and Stanley depth have the same values and can be given in terms of $k$ and $n$. For $n\geq 2k+1$, let $n\equiv 0,k+1,k+2,\dots, n-1(\mod(2k+1))$. Then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the $k^{th}$ power of a cycle on $n$ vertices and tight bounds otherwise, in terms of $n$ and $k$. We also compute lower bounds for the Stanley depth of the edge id…
We derive two general bounds for the depths of powers of squarefree monomial ideals corresponding to hyperforests. These bounds generalize known bounds for the depths of squarefree monomial ideals, which were given in terms of the edgewise domination number of the corresponding hypergraphs and the lengths of initially regular sequences with respect to the ideals.
We give tight bounds for the Stanley depth of the quotient ring of the path ideal of a cycle graph. In particular, we prove that it satisfies the Stanley inequality.
We consider the path ideal associated to a line graph, we compute \texttt{sdepth} for its quotient ring and note that it is equal with its \texttt{depth}. In particular, it satisfies the Stanley inequality.
We give upper bounds for the Stanley depth of edge ideals of certain k-partite clutters. In particular, we generalize a result of Ishaq about the Stanley depth of the edge ideal of a complete bipartite graph. A result of Pournaki, Seyed Fakhari and Yassemi implies that the Stanley's conjecture holds for d-uniform complete d-partite clutters. Here we give a shorter and different proof of this fact.
Let S be a polynomial algebra over a field. If I is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of S/I.
We prove that the edge ideals of line and cyclic graphs and their quotient rings satisfy the Stanley conjecture. We compute the Stanley depth for the quotient ring of the edge ideal associated to a cycle graph of length $n$, given a precise formula for $n\equiv 0,2 \pmod{3}$ and tight bounds for $n\equiv 1 \pmod{3}$. Also, we give bounds for the Stanley depth of a quotient of two monomial ideals, in combinatorial terms.
The Lyubeznik size of a monomial ideal $I$ of a polynomial ring $S$ is a lower bound for the Stanley depth of $I$ decreased by $1$. A proof given by Herzog-Popescu-Vladoiu had a gap which is solved here.
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 show
Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$ is either generated by four squarefree monomials of degrees $d$ and others of degrees $\geq d+1$, or by five special monomials of degrees $d$. If the Stanley depth of $I/J$ is $\leq d+1$ then the usual depth of $I/J$ is $\leq d+1$ too.
Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S = K[x_1, \ldots, x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that, for each of the integers $0 \leq r \leq d$, there exists a monomial order $<_r$ on $S$ with ${\rm depth} (S/{\rm in}_{<_r}(I)) = r$, where ${\rm in}_{<_r}(I)$ is the initial ideal of $I$ with respect to $<_r$.
In a recent work, Kaiser, Stehlík and Škrekovski provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs, and thus give counterexamples to a conjecture of Francisco, Ha and Van Tuyl. The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties. In this paper, we show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties.
For an arbitrary ideal $I$ in a polynomial ring $R$ we define the notion of initially regular sequences on $R/I$. These sequences share properties with regular sequences. In particular, the length of an initially regular sequence provides a lower bound for the depth of $R/I$. Using combinatorial information from the initial ideal of $I$ we construct sequences of linear polynomials that form initially regular sequences on $R/I$. We identify situations where initially regular sequences are also regular sequences, and we show that our results can be combined with polarization to improve known dep…
Let $I$ be a monomial ideal $I$ in a polynomial ring $R = k[x_1,...,x_r]$. In this paper we give an upper bound on $\overline{\dstab} (I)$ in terms of $r$ and the maximal generating degree $d(I)$ of $I$ such that $\depth R/\overline{I^n}$ is constant for all $n\geqslant \overline{\dstab}(I)$. As an application, we classify the class of monomial ideals $I$ such that $\overline{I^n}$ is Cohen-Macaulay for some integer $n\gg 0$.
Let $I_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n)$ be the $m$-path ideal of the path graph of length $n$, in the ring $S=K[x_1,\ldots,x_n]$. We prove that: $$\mathtt{depth}(S/I_{n,m}^t)=\begin{cases} n-t+2 - \left\lfloor \frac{n-t+2}{m+1} \right\rfloor - \left\lceil \frac{n-t+2}{m+1} \right\rceil, & t \leq n+1-m \\ m-1,& t > n+1-m \end{cases},\text{ for all }t\geq 1.$$ Also, we prove that $\mathtt{depth}(S/I_{n,m}) \geq \mathtt{sdepth}(S/I_{n,m}^t) \geq \mathtt{depth}(S/I_{n,m}^t)$ and $\mathtt{sdepth}(I_{n,m}^t)\geq \mathtt{depth}(I_{n,m}^t)$, for all $…
