Narrow Results

We prove that ${\beta }_p(I(G))={\beta }_{p,p+r}(I(G))$ for any staircase skew Ferrers graph $G$, where $p:=\mathrm{\text{pd}}(I(G))$ and $r:=\mathrm{\text{reg}}(I(G))$. As a consequence, Ene et al. conjecture is confirmed to hold true for the Betti numbers in the last column of the Betti table in a particular case. An explicit formula for the unique extremal Betti number of the binomial edge ideal of some closed graphs is also given.

Let $S=K[x_1,\dots ,x_n]$ be the standard graded polynomial ring, with K a field, and let $t=(t_1,\dots ,t_{d-1})\in {\mathbb{Z}}_{\ge 0}^{d-1}$, $d\ge 2$, be a $(d-1)$-tuple whose entries are non-negative integers. To a t-spread ideal I in S, we associate a unique $f_t$-vector and we prove that if I is t-spread strongly stable, then there exists a unique t-spread lex ideal which shares the same $f_t$-vector of Ivia the combinatorics of the t-spread shadows of special sets of monomials of S. Moreover, we characterize the possible $f_t$-vectors of t-vector spread strongly stable ideals generalizing the well-known theorems of Macaulay and Kruskal–Katona. Finally, we prove that among all t-spread strongly stable ideals with the same $f_t$-vector, the t-spread lex ideals have the largest Betti numbers.

An algorithm for automatic reducing of the topology of a h-genus three-dimensional polyhedron to the topology of a ball is proposed. The polyhedron is assumed to be represented by a three-dimensional cell complex. A special technique for separation of two-dimensional subcomplexes producing a cutting surfaces is described. This technique is based on aggregation of cells and calculation of Betti groups of the polyhedron.

A path ideal of a tree is an ideal whose minimal generating set corresponds to paths of a specified length in a tree. We provide a description of a collection of induced subtrees whose vertex sets correspond to the multi-graded Betti numbers on the linear strand in the corresponding minimal free resolution of the path ideal. For two classes of path ideals, we give an explicit description of a collection of induced subforests whose vertex sets correspond to the multi-graded Betti numbers in the corresponding minimal free resolutions. Lastly, in both classes of path ideals considered, the graded Betti numbers are explicitly computed for $t$-ary trees.

We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of paths and cycles. As a consequence, we can give new and short proofs for the known formulas of regularity and projective dimensions of path ideals of paths.

For an (oriented) graph $G$ on the vertex set $\{0,1,\dots ,n\}$ (rooted at $0$), Postnikov and Shapiro (Trans. Amer. Math. Soc.356 (2004) 3109–3142) associated a monomial ideal ${{ℳ}}_G$ in the polynomial ring $R=k[x_1,\dots ,x_n]$ over a field $k$ such that the number of standard monomials of $\frac{R}{{{ℳ}}_G}$ equals the number of (oriented) spanning trees of $G$ and hence, ${dim}_k(\frac{R}{{{ℳ}}_G})=det(L_G)$, where $L_G$ is the truncated Laplace matrix of $G$. The standard monomials of $\frac{R}{{{ℳ}}_G}$ correspond bijectively to the $G$-parking functions. In this paper, we study a monomial ideal $J_n$ in $R$ having rich combinatorial properties. We show that the minimal free resolution of the monomial ideal $J_n$ is the cellular resolution supported on a subcomplex of the first barycentric subdivision $Bd({\mathrm{\Delta }}_{n-1})$ of an $n-1$ simplex ${\mathrm{\Delta }}_{n-1}$. The integer sequence ${\{{dim}_k(\frac{R}{J_n})\}}_{n\ge 1}$ has many interesting properties. In particular, we obtain a formula, ${dim}_k(\frac{R}{J_n})=det({[m_{ij}]}_{n\times n})$, with $m_{ij}=1$ for $i>j$, $m_{ii}=i$ and $m_{ij}=i-j$ for $i<j$, similar to ${dim}_k(\frac{R}{{{ℳ}}_G})=det(L_G)$.

In this paper, we study the generalized path ideals, which is a new class of path ideals of cycle graphs. These ideals naturally generalize the standard path ideals of cycles, as studied by Alilooee and Faridi [On the resolution of path ideals of cycles, Comm. Algebra43 (2015) 5413–5433]. We give some formulas to compute all the top degree graded Betti numbers of these path ideals of cycle graphs. As a consequence, we can give some formulas to compute their projective dimension and regularity.

We introduce a squarefree monomial ideal associated to the set of domino tilings of a $2\times n$ rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo–Mumford regularity of the ideal.

Bresinsky defined a class of monomial curves in ${𝔸}^4$ with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behavior of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for the defining ideal of this class of curves.

In this paper, we compute the graded Betti numbers of toric algebras of certain bipartite graphs $G_{r,s,d}$. Also, the Castelnuovo–Mumford regularity, Hilbert series and multiplicity of $K[G_{r,s,d}]$ are explicitly determined.

Let $I$ be an ideal of the form

We express the multigraded Betti numbers of monomial ideals in four variables in terms of the multigraded Betti numbers of 66 squarefree monomial ideals, also in four variables. We use this class of 66 ideals to prove that monomial resolutions in four variables are independent of the base field. In addition, we give a formula for the Betti numbers of an arbitrary monomial ideal in four variables.

Let Δ be a stable simplicial complex on n vertexes. Over an arbitrary base field K, the symmetric algebraic shifted complex Δ^s of Δ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in the polynomial ring K[x₁, x₂, …, x_n] of the symmetric algebraic shifted complex, exterior algebraic shifted complex and combinatorial shifted complex of Δ are equal.

Let R be a commutative Noetherian local ring, 𝔞 an ideal of R, and M a finitely generated generalized f-module. Let t be a positive integer such that and t > dim M - dim M/𝔞M. In this paper, we prove that there exists an ideal 𝔟 ⊇ 𝔞 such that (1) dim M - dim M/𝔟M = t; and (2) the natural homomorphism is an isomorphism for all i > t and it is surjective for i = t. Also, we show that if is a finite set for all i < t, then there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and for all i < t.

We look for complete intersections containing certain arithmetically Cohen-Macaulay schemes, and give a complete description in the case of 2-codimensional arithmetically Cohen-Macaulay schemes and 3-codimensional arithmetically Gorenstein schemes. In particular, we prove that in these cases the sets of types of complete intersections containing such schemes have a unique minimal element and we compute it.

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 change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal. Using a Gröbner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervaire resolution for stable monomial ideals, the Betti numbers of this initial ideal is computed and it is proved that for some significant values of t, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper naturally arise in the study of generic singularities of algebraic varieties.

Let S=K[x₁,…,x_n] be a polynomial ring. Herzog and Zheng conjectured that the i-th Hilbert coefficient of a finitely generated graded Cohen-Macaulay S-module N generated in degree 0 is bounded by the functions of the minimal and maximal shifts in the minimal graded free resolution of N over S and the 0-th Betti number of N. Also, Römer asked whether under the Cohen-Macaulay assumption the i-th Betti number of S/I, where I ⊂ S is a graded ideal, can be bounded by the functions of the minimal and maximal shifts of S/I. In this paper, we provide elementary proofs to establish Herzog and Zheng's conjecture and the upper bound part of Römer's question for the Stanley-Reisner ring of a matroid complex.

For an ideal I of a Noetherian local ring one has . It is demonstrated that some residual intersections of an ideal I for which or 0 are perfect.

Let R₀ be a Noetherian local ring and R a standard graded R₀-algebra with maximal ideal 𝔪 and residue class field 𝕂 = R/𝔪. For a graded ideal I in R we show that for k ≫ 0: (1) the Artin-Rees number of the syzygy modules of I^k as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and (2) R is a polynomial ring over the regular R₀, the ring R/I^k is Golod, its Poincaré-Betti series is rational and the Betti numbers of the free resolution of 𝕂 over R/I^k are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of I^k for k ≫ 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/I^k over R.

We introduce first the spanning simplicial complex (SSC) of a multigraph $\mathcal{G}$, which gives a generalization of the SSC associated with a simple graph $G$. Combinatorial properties are discussed for the SSC of a family of uni-cyclic multigraphs ${\mathcal{U}}_{n,m}^r$ with $n$ edges including $r$ multiple edges within and outside the cycle of length $m$, which are then used to compute the $f$-vector and Hilbert series of face ring $k[{\mathrm{\Delta }}_s({\mathcal{U}}_{n,m}^r)]$ for the SSC${\mathrm{\Delta }}_s({\mathcal{U}}_{n,m}^r)$. Moreover, we find the associated primes of the facet ideal $I_{\mathcal{F}}({\mathrm{\Delta }}_s({\mathcal{U}}_{n,m}^r))$. Finally, we device a formula for homology groups of ${\mathrm{\Delta }}_s({\mathcal{U}}_{n,m}^r)$ and prove that the SSC of a family of uni-cyclic multigraphs is Cohen-Macaulay.