Related papers: Rigid monomial ideals
We construct the minimal resolutions of three classes of monomial ideals: dominant, 1-semidominant, and 2-semidominant ideals. The families of dominant and 1-semidominant ideals extend those of complete and almost complete intersections. We…
We introduce to the context of multigraded modules the methods of modules over categories from algebraic topology and homotopy theory. We develop the basic theory quite generally, with a view toward future applications to a wide class of…
For a monomial ideal $I$, let $G(I)$ be its minimal set of monomial generators. If there is a total order on $G(I)$ such that the corresponding Lyubeznik resolution of $I$ is a minimal free resolution of $I$, then $I$ is called a Lyubeznik…
In this paper, we study a class $\mathcal{C}$ of squarefree monomial ideals $I\subseteq R=\mathbb{K}[x_1,\dots,x_n]$ over a field $\mathbb{K}$, defined by the condition that $\dim R/I$ equals the maximum degree of the minimal generators of…
Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the…
We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important…
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…
We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except…
Squarefree monomial ideals arising from finite meet-semilattices and their free resolutions are studied. For the squarefree monomial ideals corresponding to poset ideals in a distributive lattice the Alexander dual is computed.
We introduce the concept of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and identify all the quasi-linear quadratic monomial ideals. We define a strongly linear monomial for a monomial ideal…
We study the WLP and SLP of artinian monomial ideals in $S=\mathbb{K}[x_1,\dots ,x_n]$ via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of $S/I$ is linear for at…
In this paper, we extend a result of Eisenbud-Reeves-Totaro in the frame of ideals of Borel type. As a consequence, we obtain a linear upper bound for the regularity of a new class of ideals, called $\mathcal D$-fixed ideals.
An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional…
The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of…
Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a…
Toward a partial classification of monomial ideals with $d$-linear resolution, in this paper, some classes of $d$-uniform clutters which do not have linear resolution, but every proper subclutter of them has a $d$-linear resolution, are…
This paper primarily studies monomial ideals by their associated lcm-lattices. It first introduces notions of weak coordinatizations of finite atomic lattices which have weaker hypotheses than coordinatizations and shows the…
We explore a family of monomial ideals derived as Gr\"obner degenerations of determinantal ideals. These ideals, previously examined as block diagonal matching field ideals within the realm of toric degenerations of Grassmannians, are…
We use discrete Morse theory to study free resolutions of monomial ideals in combination with splitting techniques. We establish the minimality of such pruned resolutions for several classes of ideals, including stable and linear quotient…
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…