Splittings of monomial ideals

International Electronic Journal of Algebra, 2021

Let K be a field and S = K[x 1 ,. .. , x n ] be a polynomial ring over K. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of such a class of squarefree monomial ideals. 1 Introduction Let K be a field and S = K[x 1 ,. .. , x n ] be the polynomial ring in n variables with coefficients in K. A squarefree monomial ideal of S is a monomial ideal generated by squarefree monomials. Such ideals are also known as Stanley-Reisner ideals, and quotients by them are called Stanley-Reisner rings. The combinatorial nature of these algebraic objects comes from their close connections to simplicial topology. Many authors have studied the class of squarefree monomial ideals from the viewpoint of commutative algebra and combinatorics (see, for example [2, 3, 6, 18], and the references therein). Let I be a graded ideal of S. A graded Betti number β k,k+ℓ (I) = 0 is called extremal if β i, i+j (I) = 0 for all i ≥ k, j ≥ ℓ, (i, j) = (k, ℓ) [4]. The pair (k, ℓ) is called a corner of I. If β k i ,k i +ℓ i (I) (i = 1,. .. , r) are extremal Betti numbers of a graded ideal I, then the set Corn(I) = {(k 1 , ℓ 1), (k 2 , ℓ 2),. .. , (k r , ℓ r)} will be called the corner sequence of I [7, Definition 4.1]. In the Macaulay or CoCoA Betti diagram of I, the graded Betti number β i,j (I) is plotted in column

The Electronic Journal of Combinatorics, 2013

We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs depending on induced subgraphs.

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2014

Let J G denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials x i y j − x j y i , 1 ≤ i < j ≤ n, in the polynomial ring S = K[x 1 ,. .. , x n , y 1 ,. .. , y n ] where {i, j} is an edge of G. We study the arithmetic properties of S/J G for G, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of deficiencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of S/J G .

Communications in Algebra, 2013

B. Sturmfels and S. Sullivant associated to any graph a toric ideal, called the cut ideal. We consider monomial cut ideals and we show that their algebraic properties such as the minimal primary decomposition, the property of having a linear resolution or being Cohen-Macaulay may be derived from the combinatorial structure of the graph.

Journal of Combinatorial Theory, Series A, 2013

Let I(G) be the edge ideal of a simple graph G. In this paper, we will give sufficient and necessary combinatorial conditions of G in which the second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay). As an application of our results, we will classify all bipartite graphs in which the second (symbolic) powers are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay).

2011

We give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of any homogeneous ideal in a polynomial ring over a field of characteristic 0.

Journal of Algebraic Combinatorics, 2014

Consider an ideal I ⊂ K[x1, . . . , xn], with K an arbitrary field, generated by monomials of degree two. Assuming that I does not have a linear resolution, we determine the step s of the minimal graded free resolution of I where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree s + 3, and we compute the corresponding graded Betti number βs,s+3. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.

2018

We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals: Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincar\&#39;e series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs.

In dimension two, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means to find their defining equations.

Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry, 2018

An n-crown C n,n on 2n vertices is a graph obtained from complete bipartite graph K n,n by removing edges of a perfect matching. Given a finite simple graph G, one can associate a simplicial complex (G). In this paper, we use combinatorial data from the associated simplicial complex (C n,n) of the crown graph C n,n and give a formula to find Betti numbers of the form β i,i+1 of edge ideals of C n,n. We also present a formula to find a particular Betti of the edge ideal of a crown graph. We explicitly compute the projective dimension of the edge ideals of crown graphs using domination parameters of the graphs.