Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences

Let A(C) be the coordinate ring of a monomial curve C⊆Aⁿ corresponding to the numerical semigroup S minimally generated by a sequence a₀,…,a_n. In the literature, little is known about the Betti numbers of the corresponding associated graded ring gr_m(A) with respect to the maximal ideal m of A=A(C). In this paper we characterize the numerical invariants of a minimal free resolution of gr_m(A) in the case a₀,…,a_n is a generalized arithmetic sequence.     

A study of the lex plus powers conjecture

An {a₁,…,a_n}-lex plus powers ideal is a monomial ideal in I⊂k[x₁,…,x_n] which minimally contains the regular sequence x₁^a₁,…,x_n^a_n and such that whenever m∈R_t is a minimal generator of I and m′∈R_t is greater than m in lex order, then m′∈I. Conjectures of Eisenbud et al. and Charalambous and Evans predict that after restricting to ideals containing a regular sequence in degrees {a₁,…,a_n}, then {a₁,…,a_n}-lex plus powers ideals have extremal properties similar to those of the lex ideal. That is, it is proposed that a lex plus powers ideal should give maximum possible Hilbert function growth (Eisenbud et al.), and, after fixing a Hilbert function, that the Betti numbers of a lex plus powers ideal should be uniquely largest (Charalambous, Evans). The first of these assertions would extend Macaulay's theorem on Hilbert function growth, while the second improves the Bigatti, Hulett, Pardue theorem that lex ideals have largest graded Betti numbers. In this paper we explore these two conjectures. First we give several equivalent forms of each statement. For example, we demonstrate that the conjecture for Hilbert functions is equivalent to the statement that for a given Hilbert function, lex plus powers ideals have the most minimal generators in each degree. We use this result to prove that it is enough to show that lex plus powers ideals have the most minimal generators in the highest possible degree. A similar result holds for the stronger conjecture. In this paper we also prove that if the weaker conjecture holds, then lex plus powers ideals are guaranteed to have largest socles. This suffices to show that the two conjectures are equivalent in dimension ?3, which proves the monomial case of the conjecture for Betti numbers in those degrees. In dimension 2, we prove both conjectures outright.     

The Hilbert functions which force the Weak Lefschetz Property

The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in [A. Wiebe, The Lefschetz property for componentwise linear ideals and Gotzmann ideals, Comm. Algebra 32 (12) (2004) 4601-4611]) whose Gotzmann ideals have the WLP.This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Betti numbers) have the WLP as well. However, we will answer in the negative, even in the case of level algebras, the most natural question that one might ask after reading the previous sentence: If A is an artinian algebra enjoying the WLP, do all artinian algebras with the same Hilbert function as A and Betti numbers lower than those of A have the WLP as well?Also, as a consequence of our result, we have another (simpler) proof of the fact that all codimension 2 algebras enjoy the WLP (this fact was first proven in [T. Harima, J. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for Artinian K-algebras, J. Algebra 262 (2003) 99-126], where it was shown that even the Strong Lefschetz Property holds).     

Extensions of the Multiplicity conjecture

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded -algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several directions. We discuss when these bounds are sharp, find a sharp lower bound in the case of not necessarily arithmetically Cohen-Macaulay one-dimensional schemes of 3-space, and propose an upper bound for finitely generated graded torsion modules. We establish this bound for torsion modules whose codimension is at most two.

     

Unimodal Gorenstein h-vectors without the Stanley–Iarrobino property

The study of the h-vectors of graded Gorenstein algebras is an important topic in combinatorial commutative algebra, which despite the large amount of literature produced during the last several years, still presents many interesting open questions. In this note, we commence a study of those unimodal Gorenstein h-vectors that do not satisfy the Stanley–Iarrobino property. Our main results, which are characteristic free, show that such h-vectors exist: 1) In socle degree e if and only if e≥6; and 2) in every codimension five or greater. The main case that remains open is that of codimension four, where no Gorenstein h-vector is known without the Stanley–Iarrobino property. We conclude by proposing the following very general conjecture: The existence of any arbitrary level h-vector is independent of the characteristic of the base field.     

On the Broken-Circuit Complex of Graphs

First, we construct a bijection between the set of h-vectors and the set of socle-vectors of Artinian algebras. As a corollary, we find the minimum codimension that an Artinian algebra with a given socle-vector can have. Then, we study the main problem in the article: Determining when there is a unique socle-vector for a given h-vector. We solve the problem completely if the codimension is at most 3.     

Note on bounds for multiplicities

Let S=K[x₁,…,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.     

Liftings of graded quasi-Hopf algebras with radical of prime codimension

Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p; that is, A with gr(A) in RG(p), where gr(A) is the associated graded algebra taken with respect to the radical filtration on A. The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p. Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H(2), H_±(p), A(q), or H(32), constructed in [5] and [8]. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories.     

Representations of matroids and free resolutions for multigraded modules

Let k be a field, let R=k[x₁,…,xm] be a polynomial ring with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, and let be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T_•(Φ,S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation ? over k of a matroid M a canonical finite complex of finite dimensional k-vector spaces T_•(?) that is a resolution of Ker?. We also show that the length of T_•(?) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map ?.     

The multiplicity and the Cohen-Macaulayness of extended Rees algebras of equimultiple ideals

Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T=A[It,t^-1]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I) over a Cohen-Macaulay ring in terms of reduction numbers and ideals.     

The determinantal ideals of extended Hankel matrices

In this paper, we use the tools of Gröbner bases and combinatorial secant varieties to study the determinantal ideals I_t of the extended Hankel matrices. Denote by c-chain a sequence a₁,…,a_k with a_i+c<a_i+1 for all i=1,…,k−1. Using the results of c-chain, we solve the membership problem for the symbolic powers and we compute the primary decomposition of the product I_t1?I_tk of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product I_t1?I_tk has a linear resolution and the multi-homogeneous Rees algebra is defined by a Gröbner basis of quadrics.     

Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum

We investigate simultaneous solutions of the matrix Sylvester equations A_iX-XB_i=C_i,i=1,2,…,k, where {A₁,…,A_k} and {B₁,…,B_k} are k-tuples of commuting matrices of order m×m and p×p, respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k-tuple of m×p matrices {C₁,…,C_k} if and only if the joint spectra σ(A₁,…,A_k) and σ(B₁,…,B_k) are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k-tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations.     

Perfect Skolem sets

A Skolem sequence is a sequence s₁,s₂,…,s_2n (where s_i∈A={1,…,n}), each s_i occurs exactly twice in the sequence and the two occurrences are exactly s_i positions apart. A set A that can be used to construct Skolem sequences is called a Skolem set. The problem of deciding which sets of the form A={1,…,n} are Skolem sets was solved by Thoralf Skolem in the late 1950s. We study the natural generalization where A is allowed to be any set of n positive integers. We give necessary conditions for the existence of Skolem sets of this generalized form. We conjecture these necessary conditions to be sufficient, and give computational evidence in favor of our conjecture. We investigate special cases of the conjecture and prove that the conjecture holds for some of them. We also study enumerative questions and show that this problem has strong connections with problems related to permutation displacements.     

The minimum distance of sets of points and the minimum socle degree

Let K be a field of characteristic 0. Let be a reduced finite set of points, not all contained in a hyperplane. Let be the maximum number of points of Γ contained in any hyperplane, and let . If I⊂R=K[x₀,…,x_n] is the ideal of Γ, then in Tohaˇneanu (2009) [12] it is shown that for n=2,3, d(Γ) has a lower bound expressed in terms of some shift in the graded minimal free resolution of R/I. In these notes we show that this behavior holds true in general, for any n≥2: d(Γ)≥A_n, where A_n=min{a_i−n} and ⊕_iR(−a_i) is the last module in the graded minimal free resolution of R/I. In the end we also prove that this bound is sharp for a whole class of examples due to Juan Migliore (2010) [10].     

Artinian level modules of embedding dimension two

We prove that a sequence of positive integers (h₀,h₁,…,h_c) is the Hilbert function of an artinian level module of embedding dimension two if and only if h_i−1−2h_i+h_i+1≤0 for all 0≤i≤c, where we assume that h₋₁=h_c+1=0. This generalizes a result already known for artinian level algebras. We provide two proofs, one using a deformation argument, the other a construction with monomial ideals. We also discuss liftings of artinian modules to modules of dimension one.     

On the multigraded Hilbert and Poincaré-Betti series and the Golod property of monomial rings

In this paper we study the multigraded Hilbert and Poincaré-Betti series of A=S/a, where S is the ring of polynomials in n indeterminates divided by the monomial ideal a. There is a conjecture about the multigraded Poincaré-Betti series by Charalambous and Reeves which they proved in the case where the Taylor resolution is minimal. We introduce a conjecture about the minimal A-free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves and, in addition, gives a formula for the Hilbert series. Using Algebraic Discrete Morse theory, we prove that the homology of the Koszul complex of A with respect to x₁,…,x_n is isomorphic to a graded commutative ring of polynomials over certain sets in the Taylor resolution divided by an ideal r of relations. This leads to a proof of our conjecture for some classes of algebras A. We also give an approach for the proof of our conjecture via Algebraic Discrete Morse theory in the general case.The conjecture implies that A is Golod if and only if the product (i.e. the first Massey operation) on the Koszul homology is trivial. Under the assumption of the conjecture we finally prove that a very simple purely combinatorial condition on the minimal monomial generating system of a implies Golodness for A.     

On the rigidity of the cotangent complex at the prime 2

In [D. Quillen, On the (co)homology of commutative rings, Proc. Symp. Pure Math. 17 (1970) 65-87; L. Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Annals of Math. 2 (150) (1999) 455-487] a conjecture was posed to the effect that if R→A is a homomorphism of Noetherian commutative rings then the flat dimension, as defined in the derived category of A-modules, of the associated cotangent complex L_A/R satisfies: . The aim of this paper is to initiate an approach for solving this conjecture when R has characteristic 2 using simplicial algebra techniques. To that end, we obtain two results. First, we prove that the conjecture can be reframed in terms of certain nilpotence properties for the divided square γ₂ and the André operation ? as it acts on Tor_R(A,?), ? any residue field of A. Second, we prove the conjecture is valid in two cases: when and when R is a Cohen-Macaulay ring.     

Generic initial ideals and graded Artinian-level algebras not having the Weak-Lefschetz Property

We find a sufficient condition that is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function cannot be level if h_d≤2d+3, and that there exists a level O-sequence of codimension 3 of type for h_d≥2d+k for k≥4. Furthermore, we show that is not level if , and also prove that any codimension 3 Artinian graded algebra A=R/I cannot be level if . In this case, the Hilbert function of A does not have to satisfy the condition h_d−1>h_d=h_d+1.Moreover, we show that every codimension n graded Artinian level algebra having the Weak-Lefschetz Property has a strictly unimodal Hilbert function having a growth condition on (h_d−1−h_d)≤(n−1)(h_d−h_d+1) for every d>θ where

h₀<h₁<<h_α==h_θ>>h_s−1>h_s.