相关论文: A duality theorem for syzygies of Veronese ideals
In this paper, we study the multigraded Betti numbers of Veronese embeddings of projective spaces. Due to Hochster's formula, we interpret these multigraded Betti numbers in terms of the homology of certain simplicial complexes. By…
We study the minimal free resolution of the Veronese modules of the polynomial ring in n variables, by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We characterize when…
We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij-S\"oderberg conjectures describing numerical invariants of syzygies. The new foundation allows us to extend the reach of…
We formulate several conjectures which shed light on the structure of Veronese syzygies of projective spaces. Our conjectures are based on experimental data that we derived by developing a numerical linear algebra and distributed…
We use the results by Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the…
In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded $k$-th syzygy module over the polynomial ring. If in addition the module is…
A celebrated theorem of Fr\"oberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active…
Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomial ideals. The notion of Alexander duality is…
We study various ideals arising in the theory of system reliability. We use ideas from the theory of divisors, orientations and matroids on graphs to describe the minimal polyhedral cellular free resolutions of these ideals. In each case we…
We study the syzygies of secant ideals of Veronese subrings of a fixed commutative graded algebra over a field of characteristic 0. One corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring…
In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $\Delta$ is vertex decomposable if and only if $I_{\Delta^{\vee}}$…
Well ordered covers of square-free monomial ideals are subsets of the minimal generating set ordered in a certain way that give rise to a Lyubeznik resolution for the ideal, and have guaranteed nonvanishing Betti numbers in certain degrees.…
Given a square-free monomial ideal $I$, we define a simplicial complex labeled by the generators of $I^2$ which supports a free resolution of $I^2$. As a consequence, we obtain (sharp) upper bounds on the Betti numbers of the second power…
An explicit construction is given of a minimal free resolution of the ideal generated by all squarefree monomials of a given degree. The construction relies upon and exhibits the natural action of the symmetric group on the syzygy modules.…
We study certain random simplicial complexes, called random quota complexes. A quota complex on $N+1$ weighted vertices is constructed by adding an $n$-simplex if the sum of the weights of the vertices is below a given quota, $q$. In this…
In this paper we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behaviour of the…
We express the multigraded Betti numbers of an arbitrary monomial ideal in terms of the multigraded Betti numbers of two basic classes of ideals. This decompo- sition has multiple applications. In some concrete cases, we use it to construct…
We prove a duality theorem for Cohen--Macaulay simplicial complexes. This is a generalisation of Poincar\'e Duality, framed in the language of combinatorial sheaves. Our treatment is self-contained and accessible for readers with a working…
We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals, Stanley--Reisner ideals of vertex-decomposable complexes and ideals with…
We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are…