Related papers: A Minimal Poset Resolution of Stable Ideals
We use the natural homeomorphism between a regular CW-complex X and its face poset P_X to establish a canonical isomorphism between the cellular chain complex of X and the result of applying the poset construction of arXiv:0806.4532v1 to…
In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable…
We introduce the notion of a \emph{resolution supported on a poset}. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of…
Given a monomial ideal $I$ with minimal free resolution $\mathcal{F}$ supported in characteristic $p>0$ on a CW-complex $X$ with regular $2$-skeleton, we construct a CW-complex $Y$ that also supports~$\mathcal{F}$ and such that the face…
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 prove that the Pommaret-Seiler resolution for quasi-stable ideals is cellular and give a cellular structure for it. This shows that this resolution is a generalization of the well known Eliahou-Kervaire resolution for stable ideals in a…
Let $\Bbbk$ be a field, and let $I$ be a monomial ideal in the polynomial ring $R=\Bbbk[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex that provides a finite free resolution of $R/I$ as an $R$-module. Building on this,…
We give the resolutions of co-letterplace ideals of posets in a completely explicit, very simple form. This generalizes and simplifies a number of linear resolutions in the literature, among them the Eliahou-Kervaire resolutions of strongly…
We prove that the Eliahou-Kervaire resolution of a Cohen-Macaulay stable monomial is supported by a regular CW complex whose underlying space is a closed ball. We also show that the modified Eliahou-Kervaire resolutions of variants of a…
We study when Taylor resolutions of monomial ideals are minimal. We consider monomial ideals with linear quotients. In particular, we determine precisely the stable ideals and the monomial ideals with linear resolutions having the miminal…
In this paper we investigate the class of rigid monomial ideals. We give a characterization of the minimal free resolutions of certain classes of these ideals. Specifically, we show that the ideals in a particular subclass of rigid monomial…
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…
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…
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…
We use the lcm-lattice of a monomial ideal to study its minimal free resolutions. A new concept called a Taylor basis of a minimal free resolution is introduced and then used throughout the paper. We give a method of constructing minimal…
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…
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…
The support poset of a monomial ideal $I\subseteq\mathbf{k}[x_1,\dots,x_n]$ encodes the relation between the variables $x_1,\dots,x_n$ and the minimal monomial generators of $I$. It is known that not every poset is realizable as the support…
One can iteratively obtain a free resolution of any monomial ideal $I$ by considering the mapping cone of the map of complexes associated to adding one generator at a time. Herzog and Takayama have shown that this procedure yields a minimal…
A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form…