Divisors on graphs,binomial and monomial ideals,and cellular resolutions

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their \({\mathbb {Z}}\)-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.     

On pseudo symmetric monomial curves

We study monomial curves, toric ideals and monomial algebras associated to 4-generated pseudo symmetric numerical semigroups. Namely, we determine indispensable binomials of these toric ideals, give a characterization for these monomial algebras to have strongly indispensable minimal graded free resolutions. We also characterize when the tangent cones of these monomial curves at the origin are Cohen–Macaulay.     

Cellular resolutions of ideals defined by nondegenerate simplicial homomorphisms

In this paper we introduce the class of ordered homomorphism ideals and prove that these ideals admit minimal cellular resolutions constructed as homomorphism complexes. Motivated by work of Dochtermann and Engström, we introduce the class of cointerval simplicial complexes and investigate their combinatorial and topological properties. As a concrete illustration of these structural results, we introduce and study nonnesting monomial ideals, an interesting family of combinatorially defined ideals.     

Trees, parking functions, syzygies, and deformations of monomial ideals

For a graph , we construct two algebras whose dimensions are both equal to the number of spanning trees of . One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to -parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

     

Regularity,depth and arithmetic rank of bipartite edge ideals

We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.     

A characterization of non-removable ideals in commutative locally multiplicatively pseudoconvex algebras

We introduce here the concept of weak joint topological divisors of zero in commutative locally multiplicatively pseudoconvex Hausdorff algebras and use it for the characterization of non-removable ideals in these algebras. Examples of removable and non-removable ideals in these algebras are given.     

Jet schemes and minimal toric embedded resolutions of rational double point singularities

Using the structure of the jet schemes of rational double point singularities, we construct “minimal embedded toric resolutions” of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every “minimal embedded toric resolution”. We prove that this correspondence is bijective except for the E₈ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.     

Free resolutions of the defining ideal of certain rational surfaces

We consider the surface obtained from the projective plane by blowing up the points of intersection of two plane curves of same degree meeting transversely. We find minimal free resolutions of the defining ideals of these surfaces embedded in projective space by the sections of a very ample divisor class. All of the results are proven over an algebraically closed field of arbitrary characteristic.     

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

Structure of Matrices and Their Divisors over the Domain of Principal Ideals

We investigate the structure of matrices and their divisors over the domain of principal ideals.     

Singularities of the structure of two-sided ideals of a domain of elementary divisors

We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is equal to zero. We also show that a Bézout domain with finitely many two-sided ideals is a domain of elementary divisors if and only if it is a 2-simple Bézout domain.     

Minimal free resolutions that are not supported by a CW-complex

We provide the first examples of minimal free resolutions of monomial ideals which cannot be supported by any CW-complex. Their existence answers negatively the fundamental question of whether every monomial ideal admits a minimal cellular resolution.     

Betti-linear ideals

We introduce the notion of a Betti-linear monomial ideal, which generalizes the notion of lattice-linear monomial ideal introduced by Clark. We provide a characterization of Betti-linearity in terms of Tchernev's poset construction. As an application we obtain an explicit canonical construction for the minimal free resolutions of monomial ideals having pure resolutions.     

Structure of rings of functions with Riemann Stieltjes convolution products

In this note three sets of complex valued functions with pointwise addition and a Riemann Stieltjes convolution product are considered. The functions considered are discrete analytic functions, sequences, and continuous functions of bounded variation defined on the nonnegative real numbers. Each forms a commutative algebra with identity. The discrete analytic functions form a principal ideal ring with five maximal ideals, nine prime ideals, and is essentially a direct sum of four discrete valuation rings. The ring of sequences is isomorphic to an ideal of the ring of discrete analytic functions; it has two maximal and three prime ideals. Both contain divisors of zero. The units, associates, irreducible elements and primes in these two rings are described. The results are used to study the continuous functions; partial results are obtained concerning units and divisors of zero. The product satisfies a convolution theorem.     

The depth of the Jacobian ring of a homogeneous polynomial in three variables

The question as to whether the Jacobian ideal of an irreducible projective plane curve always admits an irrelevant component has been going around for some years. One shows that a curve will satisfy this if it has only ordinary nodes or cusps, while an example is given of a family of sextic curves whose respective Jacobian ideals are saturated. The connection between this problem and the theory of homogeneous free divisors in three variables is also pointed out, so the example gives a family of Koszul-free divisors.

     

Pseudocomplementation and minimal prime ideals in semirings

In Section 2 of the present paper, we introduce the concept of pseudocomplementation for semirings and show the semiring version of some known results in lattice theory. We also introduce semirings with pc-functions and prove some interesting results for minimal prime ideals of such semirings. In Section 3, some classical results for minimal prime ideals in ring theory are generalized in the context of semiring theory.