Linear Quotients Research Articles

The eventual shape of the Betti table of 𝔪kM

Let [Formula: see text] be a standard graded polynomial ring over a field [Formula: see text] in a finite set of variables, and let [Formula: see text] be the graded maximal ideal of [Formula: see text]. It is known that for a finitely generated graded [Formula: see text]-module [Formula: see text] and all integers [Formula: see text], the module [Formula: see text] is componentwise linear. For large [Formula: see text] we describe the pattern of the Betti table of [Formula: see text] when [Formula: see text]. Moreover, we show that for any [Formula: see text], [Formula: see text] has linear quotients if [Formula: see text] is a monomial ideal.

Regularity of powers of path ideals of line graphs

Let I be the t-path ideal of Ln , the line graph with n-vertices. It is shown that Is has a linear resolution for some s ≥ 1 (or equivalently for all s ≥ 1 ) if and only if Is has linear quotients for some s ≥ 1 (or equivalently for all s ≥ 1 ) if and only if t ≤ n ≤ 2 t . In addition, we present an explicit formula for the regularity of Is for all s ≥ 1 . It turns out it is linear in s from the very beginning.

Almost isotropy-maximal manifolds of non-negative curvature

We extend the equivariant classification results of Escher and Searle for closed, simply connected, Riemannian n n -manifolds with non-negative sectional curvature admitting isometric isotropy-maximal torus actions to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In particular, we prove that any such manifold is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to three.

Mapping cones of monomial ideals over exterior algebras

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the differentials in F when the ideal I has a regular decomposition function. Finally, we get a formula for the graded Betti numbers of a new class of monomial ideals including the class of strongly stable ideals.

Rees algebra of maximal order Pfaffians and its diagonal subalgebras

Given a skew-symmetric matrix X, the Pfaffian of X is defined as the square root of the determinant of X. In this article, we give the explicit defining equations of the Rees algebra of a Pfaffian ideal I generated by the maximal order Pfaffians of a generic skew-symmetric matrix. We further prove that all diagonal subalgebras of the corresponding Rees algebra of I are Koszul. We also look at Rees algebras of Pfaffian ideals of linear type associated with certain sparse skew-symmetric matrices. In particular, we consider the tridiagonal matrices and identify the corresponding Pfaffian ideals to be of Gröbner linear type and as the vertex cover ideals of unmixed bipartite graphs. As an application of our results, we conclude that all their ordinary and symbolic powers have linear quotients.

A Characterization of Sequentially Cohen–Macaulay Matroidal Ideals

Let [Formula: see text] be the polynomial ring in [Formula: see text] variables over a field [Formula: see text] and [Formula: see text] be a matroidal ideal of [Formula: see text]. We show that [Formula: see text] is sequentially Cohen–Macaulay if and only if the [Formula: see text] has linear quotients. As a consequence, [Formula: see text] is sequentially Cohen–Macaulay if and only if [Formula: see text] is shellable.

Very well–covered graphs by Betti splittings

A very well–covered graph is an unmixed graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. We study these graphs by means of Betti splittings and mapping cone constructions. We show that the cover ideals of Cohen–Macaulay very well–covered graphs are splittable. As a consequence, we compute explicitly the minimal graded free resolution of the cover ideals of such a class of graphs and prove that these graphs have homological linear quotients. Finally, we conjecture the same is true for each power of the cover ideal of a Cohen–Macaulay very well–covered graph, and settle it in the bipartite case.

Dirac’s Theorem and Multigraded Syzygies

Let G be a simple finite graph. A famous theorem of Dirac says that G is chordal if and only if G admits a perfect elimination order. It is known by Fröberg that the edge ideal I(G) of G has a linear resolution if and only if the complementary graph $$G^c$$ of G is chordal. In this article, we discuss some algebraic consequences of Dirac’s theorem in the theory of homological shift ideals of edge ideals. Recall that if I is a monomial ideal, $$HS _k(I)$$ is the monomial ideal generated by the kth multigraded shifts of I. We prove that $$HS _1(I)$$ has linear quotients, for any monomial ideal I with linear quotients generated in a single degree. For and edge ideal I(G) with linear quotients, it is not true that $$HS _k(I(G))$$ has linear quotients for all $$k\ge 0$$ . On the other hand, if $$G^c$$ is a proper interval graph or a forest, we prove that this is the case. Finally, we discuss a conjecture of Bandari, Bayati, and Herzog that predicts that if I is polymatroidal, $$HS _k(I)$$ is polymatroidal too, for all $$k\ge 0$$ . We are able to prove that this conjecture holds for all polymatroidal ideals generated in degree two.

Linear resolutions of t-spread lexsegment ideals via Betti splittings

Let [Formula: see text] be a polynomial ring in [Formula: see text] variables with coefficients over a field [Formula: see text]. A [Formula: see text]-spread lexsegment ideal [Formula: see text] of [Formula: see text] is a monomial ideal generated by a [Formula: see text]-spread lexsegment set. We determine all [Formula: see text]-spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti numbers of such a class of ideals and furthermore we characterize all incompletely [Formula: see text]-spread lexsegment ideals with linear quotients.

Ideals with Linear Quotients and Componentwise Polymatroidal Ideals

If I is a monomial ideal with linear quotients, then it has componentwise linear quotients. However, the converse of this statement is an open question. In this paper, we provide two classes of ideals for which the converse of this statement holds. First class is the componentwise polymatroidal ideals in K[x, y] and the second one is the componentwise polymatroidal ideals with strong exchange property.

Componentwise linear powers and the $x$-condition

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gröbner basis of the defining ideal $J$ of $A$ we give a condition, called the $x$-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $\mathcal{R}(I)$ of a graded ideal or the symmetric algebra $\textrm{Sym}(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.

Second Powers of Cover Ideals of Paths

We show that the second power of the cover ideal of a path graph has linear quotients. To prove our result we construct a recursively defined order on the generators of the ideal which yields linear quotients. Our construction has a natural generalization to the larger class of chordal graphs. This generalization allows us to raise some questions that are related to some open problems about powers of cover ideals of chordal graphs.

Resolution of ideals associated to subspace arrangements

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 underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a certain poset ideal associated to the underlying subspace arrangement.

Powers of edge ideals of weighted oriented graphs with linear resolutions

Let [Formula: see text] be a weighted oriented graph and [Formula: see text] denote the corresponding edge ideal. In this paper, we give a combinatorial characterization of [Formula: see text] which has a linear resolution. As a consequence, we prove that if [Formula: see text] is the edge ideal of a weighted oriented graph [Formula: see text], then [Formula: see text] has a linear resolution if and only if all powers of [Formula: see text] have a linear resolution. Also, we prove that if [Formula: see text] is a weighted oriented graph and [Formula: see text] for all [Formula: see text], then [Formula: see text] has a linear resolution if and only if all powers of [Formula: see text] have linear quotients. We provide a lower bound for the regularity of powers of edge ideals of weighted oriented graphs in terms of induced matching. Finally, we obtain a general upper bound for the regularity of edge ideals of weighted oriented graphs.

Graded Linearity of Stanley–Reisner Ring of Broken Circuit Complexes

This paper introduces two new notions of graded linear resolution and graded linear quotients, which generalize the concepts of linear resolution property and linear quotient for modules over the polynomial ring A = k[x1, …, xn]. Besides, we compare graded linearity with componentwise linearity in general. For modules minimally generated by a regular sequence in a maximal ideal of A, we show that graded linear quotients imply graded linear resolution property for the colon ideals. On the other hand, we provide specific characterizations of graded linear resolution property for the Stanley–Reisner ring of broken circuit complexes and generalize the results of Van Le and Römer, related to the decomposition of matroids into the direct sum of uniform matroids. Specifically, we show that the matroid can be stratified such that each stratum has a decomposition into uniform matroids. We also present analogs of our results for the Orlik–Terao ideal of hyperplane arrangements which are translations of the corresponding results on matroids.

Symbolic powers of generalized star configurations of hypersurfaces

We introduce the class of sparse symmetric shifted monomial ideals. These ideals have linear quotients and their Betti numbers are computed. Using this, we prove that the symbolic powers of the generalized star configuration ideal are sequentially Cohen–Macaulay under some mild genericness assumption. With respect to these symbolic powers, we also consider the Harbourne–Huneke containment problem and establish the Demailly-like bound.

Albanese Maps and Fundamental Groups of Varieties With Many Rational Points Over Function Fields

Abstract We investigate properties of the Albanese map and the fundamental group of a complex projective variety with many rational points over some function field and prove that every linear quotient of the fundamental group of such a variety is virtually abelian, as well as that its Albanese map is surjective, has connected fibres, and has no multiple fibres in codimension one.

Torus actions, maximality, and non-negative curvature

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

A Construction of Sequentially Cohen–Macaulay Graphs

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

English

for each squarefree monomial ideal I ⊂ S = k[x1, …, xn], we associate a simple finite graph GI by using the first linear syzygies of I. The nodes of GI are the generators of I, and two vertices ui and uj are adjacent if there exist variables x, y such that xui = yuj. In the cases, where GI is a cycle or a tree, we show that I has a linear resolution if and only if I has linear quotients and if and only if I is variable-decomposable. In addition, with the same assumption on GI, we characterize all squarefree monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension 2 monomial ideals with a linear resolution. As another application of our results, we also characterize all Cohen-Macaulay simplicial complexes in the case, where $${G_{\rm{\Delta }}} \cong {G_{{I_{\rm{\Delta }}} \vee }}$$ is a cycle or a tree.