Class Of Monomial Ideals Research Articles

Results on the behavior of the Ratliff–Rush operation and the depth of the associated graded ring

We give a full characterization of the unique Ratliff–Rush complete ideals in the polynomial rings R with two variables over a field. For a class of monomial ideals in R, we investigate the Ratliff–Rush behavior of powers of these ideals and also the depth of their associated graded ring. Namely, we give some sufficient and some necessary conditions for this depth to be positive. The results of this paper allow us to answer several questions of [13].

Regularity and Koszul property of symbolic powers of monomial ideals

Let $I$ be a homogeneous ideal in a polynomial ring over a field. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. Motivated by results about ordinary powers of $I$, we study the asymptotic behavior of the regularity function $\text{reg}~ (I^{(n)})$ and the maximal generating degree function $\omega(I^{(n)})$, when $I$ is a monomial ideal. It is known that both functions are eventually quasi-linear. We show that, in addition, the sequences $\{\text{reg}~ I^{(n)}/n\}_n$ and $\{\omega(I^{(n)})/n\}_n$ converge to the same limit, which can be described combinatorially. We construct an example of an equidimensional, height two squarefree monomial ideal $I$ for which $\omega(I^{(n)})$ and $\text{reg}~ (I^{(n)})$ are not eventually linear functions. For the last goal, we introduce a new method for establishing the componentwise linearity of ideals. This method allows us to identify a new class of monomial ideals whose symbolic powers are componentwise linear.

Betti numbers of symmetric shifted ideals

We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.

The Jacobian Dual of Certain Mixed Product Ideals

We consider the symmetric algebra of a class of monomial ideals generated by s-sequences. For these ideals with linear syzygies, we determine their Jacobian dual modules and study their duality properties.

Fibonacci numbers and resolutions of domino ideals

This paper considers a class of monomial ideals, called domino ideals, whose generating sets correspond to the sets of domino tilings of a $2\times n$ tableau. The multi-graded Betti numbers are shown to be in one-to-one correspondence with equivalence classes of sets of tilings. It is well-known that the number of domino tilings of a $2\times n$ tableau is given by a Fibonacci number. Using the bijection, this relationship is further expanded to show the relationship between the Fibonacci numbers and the graded Betti numbers of the corresponding domino ideal.

ON THE STANLEY DEPTH AND SIZE OF MONOMIAL IDEALS

AbstractLet $\mathbb{K}$ be a field and S = ${\mathbb{K}}$[x1, . . ., xn] be the polynomial ring in n variables over the field $\mathbb{K}$. For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.

Minimal resolutions of dominant and semidominant ideals

We introduce new classes of monomial ideals: dominant, p-semidominant, and GNP ideals. The families of dominant and 1-semidominant ideals extend those of complete and almost complete intersections, while the family of GNP ideals extends that of generic ideals. We show that dominant ideals give a complete characterization of when the Taylor resolution is minimal, 1-semidominant ideals are Scarf, and the minimal resolutions of 2-semidominant ideals can be obtained from their Taylor resolutions by eliminating faces and facets of equal multidegree, in arbitrary order. We also generalize a theorem of Bayer, Peeva and Sturmfels by proving that GNP ideals are Scarf.

Stanley depth of weakly 0-decomposable ideals

Recently, Seyed Fakhari proved that if I is a weakly polymatroidal monomial ideal in \({S\,=\,\mathbb{K}[x_1,\ldots,x_n]}\), then Stanley’s conjecture holds for S/I, namely, sdepth \({(S/I)\,\geq\, {\rm depth}(S/I)}\). We generalize his ideas and introduce several new classes of monomial ideals which also share this property. In particular, if I is the Stanley–Reisner ideal of the Alexander dual of a nonpure vertex decomposable simplicial complex, then Stanley’s conjecture holds for S/I.

Upper and Lower Bounds for the Stanley Depth of Certain Classes of Monomial Ideals and Their Residue Class Rings

We give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We also give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph.

ON A CLASS OF MONOMIAL IDEALS

AbstractLet $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i\leq ~k\} - \mathrm{ht} (I)$, for all $n\geq 1$. In addition we show that if $I$ is MHC and $w$ is an indeterminate which is not in the monomials generating $I$, then $\mathrm{reg} (S/ \mathop{(I+ {w}^{d} S)}\nolimits ^{n} )\leq \mathrm{reg} (S/ I)+ nd- 1$ for all $n\geq 1$ and $d$ large enough. Finally, we implement an algorithm for the computation of $m(I)$.

Powers of lexsegment ideals with linear resolution

All powers of lexsegment ideals with linear resolution (equivalently, with linear quotients) have linear quotients with respect to suitable orders of the minimal monomial generators. For a large subclass of lexsegment ideals, the corresponding Rees algebra has a quadratic Grobner basis, thus it is Koszul. We also find other classes of monomial ideals with linear quotients whose powers have linear quotients too.

Prime Filtrations and Primary Decompositions of Modules

We give a new characterization for prime filtrations of an R-module M in terms of primary decomposition of the zero submodule of M. Using this characterization we prove that some classes of monomial ideals like generic and cogeneric monomial ideals are clean, pretty clean, or almost clean.

Poset resolutions and lattice-linear monomial ideals

We introduce the class of lattice-linear monomial ideals and use the lcm-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals with a linear free resolution and the class of Scarf monomial ideals. Our main tool is a new construction by Tchernev that produces from a map of posets η : P → N n a sequence of multigraded modules and maps.

Bounding multiplicity by shifts in the Taylor resolution

A weaker form of the multiplicity conjecture of Herzog, Huneke, and Srinivasan is proven for two classes of monomial ideals: quadratic monomial ideals and squarefree monomial ideals with sufficiently many variables relative to the Krull dimension. It is also shown that tensor products, as well as Stanley-Reisner ideals of certain unions, satisfy the multiplicity conjecture if all the components do. Conditions under which the bounds are achieved are also studied.