Related papers: Betti splitting via componentwise linear ideals

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a…

A certain squarefree monomial ideal $H_P$ arising from a finite partially ordered set $P$ will be studied from viewpoints of both commutative algebra and combinatorics. First, it is proved that the defining ideal of the Rees algebra of…

The ideals generated by fold products of linear forms are generalizations of powers of defining ideals of star configurations, or of Veronese type ideals, and in this paper we study their Betti numbers. In earlier work, the authors together…

In this note we consider monoidal complexes and their associated algebras, called toric face rings. These rings generalize Stanley-Reisner rings and affine monoid algebras. We compute initial ideals of the presentation ideal of a toric face…

Mats Boij and Jonas Soederberg (math.AC/0611081) have conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring can be decomposed in a certain way as a positive linear combination of Betti tables of modules with…

We study ideals generated by $n+1$ powers of general linear forms in $R= k[x_1,\dots,x_n]$. By generalizing the ideas in a recent paper of Diethorn et al., we determine the Betti numbers of such ideals when at least one generator is a…

Let $J\subset I$ be monomial ideals. We show that the Stanley depth of $I/J$ can be computed in a finite number of steps. We also introduce the $\fdepth$ of a monomial ideal which is defined in terms of prime filtrations and show that it…

When a monomial ideal has linear quotients with respect to an admissible order of increasing support-degree, we provide two proofs of different flavors to show that it is componentwise support-linear. We also introduce the variable…

The aim of this thesis is to investigate the Betti diagrams of squarefree monomial ideals in polynomial rings. We use two key tools to help us study these diagrams. The first is the Stanley-Reisner Correspondence, which assigns a unique…

Motivated by the fact that as the number of generators of an ideal grows so does the complexity of calculating relations among the generators, this paper identifies collections of monomial ideals with a growing number of generators which…

Let $M$ be a finitely generated module over a local ring $(R,\mathfrak{m})$. By $\mathcal{S}_j(M)$, we denote the $j$th symmetric power of $M$ ($j$th graded component of the symmetric algebra $\mathcal{S}_R(M)$). The purpose of this paper…

We produce a family of complexes called trimming complexes and explore applications. We study how trimming complexes can be used to deduce the Betti table for the minimal free resolution of the ideal generated by subsets of a generating set…

Let $G$ be a chordal graph and $I(G)$ its edge ideal. Let $\beta (I(G)) = (\beta_0, \beta_1, ..., \beta_p)$ denote the Betti sequence of $I(G)$, where $\beta_i$ stands for the $i$th total Betti number of $I(G)$ and where $p$ is the…

For a simplicial complex $\Delta$, the affect of the expansion functor on combinatorial properties of $\Delta$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the…

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…

Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is…

We introduce a construction, called linearization, that associates to any monomial ideal $I$ an ideal $\mathrm{Lin}(I)$ in a larger polynomial ring. The main feature of this construction is that the new ideal $\mathrm{Lin}(I)$ has linear…

Given arbitrary monomial ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $K$, we investigate the Stanley depth of powers of the sum $I+J$, and their quotient rings, in $A\otimes_K B$ in terms of those of $I$ and $J$. Our…

Boij-S\"oderberg theory shows that the Betti table of a graded module can be written as a liner combination of pure diagrams with integer coefficients. Using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these…

We give several bounds for $sdepth_S(I+J)$, $sdepth_S(I\cap J)$, $sdepth_S(S/(I+J))$, $sdepth_S(S/(I\cap J))$, $sdepth_S(I:J)$ and $sdepth_S(S/(I:J))$ where $I,J\subset S=K[x_1,...,x_n]$ are monomial ideals. Also, we give several equivalent…