Related papers: On monomial ideals whose Lyubeznik resolution is m…
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 the lexsegment ideals the…
We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the…
For an ideal $I$ in a polynomial ring over a field, a monomial support of $I$ is the set of monomials that appear as terms in a set of minimal generators of $I$. Craig Huneke asked whether the size of a monomial support is a bound for the…
Taylor presented an explicit resolution for arbitrary monomial ideals. Later, Lyubeznik found that already a subcomplex defines a resolution. We show that the Taylor resolution may be obtained by repeated application of the Schreyer Theorem…
Call a monomial ideal M "generic" if no variable appears with the same nonzero exponent in two distinct monomial generators. Using a convex polytope first studied by Scarf, we obtain a minimal free resolution of M. Any monomial ideal M can…
Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…
In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Groebner basis theory, the concept of…
In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial…
We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in S=k[x_1,x_2,...,x_n]; this includes the case of…
Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In two previous papers we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic…
Let $S$ be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of $S$ having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if Stanley's…
We show that any lexsegment ideal with linear resolution has linear quotients with respect to a suitable ordering of its minimal monomial generators. For completely lexsegment ideals with linear resolution we show that the decomposition…
This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal $I$ in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound $\mu (I^2)\ge 9$ for the number of minimal…
The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see,…
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…
We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a…
Let $I$ be a monomial ideal in a polynomial ring $A=K[x_1,...,x_n]$. We call a monomial ideal $J$ to be a minimal monomial reduction ideal of $I$ if there exists no proper monomial ideal $L \subset J$ such that $L$ is a reduction ideal of…
We prove that monomial ideals with at most five generators and their Artinian reductions have minimal generalized Barile-Macchia resolutions. As a corollary, these ideals have minimal cellular resolutions, extending a result by Faridi, D.G,…
It is known that for a monomial ideal $I$, the number of minimal generators, $\mu(I^n)$, eventually follows a polynomial pattern for increasing $n$. In general, little is known about the power at which this pattern emerges. Even less is…
The Koszul homology of modules of the polynomial ring $R$ is a central object in commutative algebra.It is strongly related with the minimal free resolution of these modules, and thus with regularity, Hilbert functions, etc. Here we…