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.     

Combinatorial bounds on Hilbert functions of fat points in projective space

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in , in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N=2, we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defined upper and lower bounds on the graded Betti numbers for the ideal I_A defining A, generalizing results of Geramita et al. (2006) [16]. We obtain the exact Hilbert functions and graded Betti numbers for many families of examples, interesting combinatorially, geometrically, and algebraically. Our method works in any characteristic.     

Fonction asymptotique de Samuel des sections hyperplanes et multiplicité

Let (A,m_A,k) be a local noetherian ring and I an m_A-primary ideal. The asymptotic Samuel function (with respect to I) : A?R∪{+∞} is defined by , ∀x∈A. Similarly, one defines, for another ideal J, as the minimum of as x varies in J. Of special interest is the rational number . We study the behavior of the asymptotic Samuel function (with respect to I) when passing to hyperplane sections of A as one does for the theory of mixed multiplicities.     

Looking out for stable syzygy bundles

We study (slope-)stability properties of syzygy bundles on a projective space PN given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.     

Strongly irreducible ideals of a commutative ring

An ideal I of a ring R is said to be strongly irreducible if for ideals J and K of R, the inclusion J∩K⊆I implies that either J⊆I or K⊆I. The relationship among the families of irreducible ideals, strongly irreducible ideals, and prime ideals of a commutative ring R is considered, and a characterization is given of the Noetherian rings which contain a non-prime strongly irreducible ideal.     

A bound for the Castelnuovo-Mumford regularity of log canonical varieties

In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal I in terms of the degrees of its generators. We assume that I defines a local complete intersection with log canonical singularities.     

Essential and inessential elements of a standard basis

In this paper we introduce the concept of inessential element of a standard basis B(I), where I is any homogeneous ideal of a polynomial ring. An inessential element is, roughly speaking, a form of B(I) whose omission produces an ideal having the same saturation as I; it becomes useless in any dehomogenization of I with respect to a linear form. We study the properties of B(I) linked to the presence of inessential elements and give some examples.     

On the multiplicity and the Cohen-Macaulayness of fiber cones of graded algebras

Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J. Let S=?_n≥0S_n be a finitely generated standard graded algebra over A. Set S⁺=?_n>0S_n. Denote by F_J(S)=?_n≥0→(S_n/JS_n) the fiber cone of S with respect to J. The paper characterizes the multiplicity and the Cohen-Macaulayness of F_J(S) in terms of minimal reductions of S⁺.     

A numerical characterization of the S2-ification of a Rees algebra

Let A be a local ring with maximal ideal . For an arbitrary ideal I of A, we define the generalized Hilbert coefficients . When the ideal I is -primary, j_k(I)=(0,…,0,(−1)^ke_k(I)), where e_k(I) is the classical kth Hilbert coefficient of I. Using these coefficients we give a numerical characterization of the homogeneous components of the S₂-ification of S=A[It,t⁻¹], extending previous results obtained by the author to not necessarily -primary ideals.     

Poincaré series of modules over compressed Gorenstein local rings

Given two positive integers e and s we consider Gorenstein Artinian local rings R   whose maximal ideal m$\mathfrak{m}$ satisfies m^s≠0=m^s+1${\mathfrak{m}}^s\ne 0={\mathfrak{m}}^{s+1}$ and _rankR/m(m/m²)=e${\mathrm{rank}}_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)=e$. We say that R is a compressed Gorenstein local ring   when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If s≠3$s\ne 3$, we prove that the Poincaré series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s  . Note that for s=3$s=3$ examples of compressed Gorenstein local rings with transcendental Poincaré series exist, due to Bøgvad.     

Depths and Cohen–Macaulay properties of path ideals

Given a tree T on n vertices, there is an associated ideal I   of R[_x1,…,_xn]$R[x_1,\dots ,x_n]$ generated by all paths of a fixed length ? of T  . We classify all trees for which R/I$R/I$ is Cohen–Macaulay, and we show that an ideal I whose generators correspond to any collection of subtrees of T satisfies the König property. Since the edge ideal of a simplicial tree has this form, this generalizes a result of Faridi. Moreover, every square-free monomial ideal can be represented (non-uniquely) as a subtree ideal of a graph, so this construction provides a new combinatorial tool for studying square-free monomial ideals.     

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.     

Intersections of Magnus subgroups and embedding theorems for cyclically presented groups

It is now known that the intersection of two Magnus subgroups M_i=〈Y_i〉 (1≤i≤2) in a one-relator group is either the free group F on Y₁∩Y₂ or the free product of F together with an infinite cyclic group (so-called exceptional intersection). Using this, we give conditions under which two embedding theorems for cyclically presented groups can be obtained. This provides a new method for proving such groups infinite. We also give a combinatorial method for checking the presence of exceptional intersections.     

The Hilbert function of a maximal Cohen–Macaulay module. Part II

Let (A,m)$(A,\mathfrak{m})$ be a strict complete intersection of positive dimension and let M be a maximal Cohen–Macaulay A-module with bounded Betti numbers. We prove that the Hilbert function of M is non-decreasing. We also prove an analogous statement for complete intersections of codimension two.     

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 ?.     

On cohomology algebras of complex subspace arrangements

The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations.

     

Graph-directed systems and self-similar measures on limit spaces of self-similar groups

Let G be a group and ?:H→G be a contracting homomorphism from a subgroup H<G of finite index. V. Nekrashevych (2005) [25] associated with the pair (G,?) the limit dynamical system (JG,s) and the limit G-space XG together with the covering _?g∈GT⋅g by the tile T. We develop the theory of self-similar measures m on these limit spaces. It is shown that (JG,s,m) is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile T has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles T∩(T⋅g) for g∈G. We present applications to the invariant measures for the rational functions on the Riemann sphere and to the evaluation of the Lebesgue measure of integral self-affine tiles.     

Filtrations and local syzygies of multiplier ideals II

Let a be a non-zero ideal sheaf on a smooth affine variety X of dimension d and let c be a positive rational number. Let x be a closed point of X and let m_x be the maximal ideal sheaf at x. In [Robert Lazarsfeld, Kyungyong Lee, Local syzygies of multiplier ideals, Invent. Math. 167 (2007) 409-418] the authors studied the local syzygies of the multiplier ideal J(a^c). Motivated by their result, the asymptotic behavior of the local syzygies of the multiplier ideal at x for k≥d−2 was studied in [Seunghun Lee, Filtrations and local syzygies of multiplier ideals, J. Algebra (2007) 629-639]. In this note, we study the local syzygies of at x for 1≤k≤d−3. As a by-product we give a different proof of the main theorem in the former reference cited above.     

Generalized Hilbert Functions

Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciuperc?. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients 𝔧₀,…, 𝔧 _d?2 are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of 𝔧 _d?1. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the ‘expected’ shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.