Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Hilbert schemes and maximal Betti numbers over veronese rings

Macaulay??s Theorem (Macaulay in Proc. Lond Math Soc 26:531?C555, 1927) characterizes the Hilbert functions of graded ideals in a polynomial ring over a field. We characterize the Hilbert functions of graded ideals in a Veronese ring R (the coordinate ring of a Veronese embedding of P ^r-1). We also prove that the Hilbert scheme, which parametrizes all graded ideals in R with a fixed Hilbert function, is connected; this is an analogue of Hartshorne??s Theorem (Hartshorne in Math. IHES 29:5?C48, 1966) that Hilbert schemes over a polynomial ring are connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the same Hilbert function.     

Artinian Level Algebras of Low Socle Degree

In this article we study Hilbert functions and isomorphism classes of Artinian level local algebras via Macaulay's inverse system. Upper and lower bounds concerning numerical functions admissible for level algebras of fixed type and socle degree are known. For each value in this range we exhibit a level local algebra with that Hilbert function, provided that the socle degree is at most three. Furthermore, we prove that level local algebras of socle degree three and maximal Hilbert function are graded. In the graded case, the extremal strata have been parametrized by Cho and Iarrobino.     

Abstract Hilbert Schemes

In analogy with classical projective algebraic geometry, Hilbert functors can be defined for objects in any Abelian category. We study the moduli problem for such objects. Using Grothendieck's general framework. We show that with suitable hypotheses the Hilbert functor is representable by an algebraic space locally of finite type over the base field. For the category of the graded modules over a strongly Noetherian graded ring, the Hilbert functor of graded modules with a fixed Hilbert series is represented by a commutative projective scheme. For the projective scheme corresponding to a suitable noncommutative graded algebra, the Hilbert functor is represented by a countable union of commutative projective schemes.     

On the Hilbert function of Artinian local complete intersections of codimension three

In singularity theory or algebraic geometry, it is natural to investigate possible Hilbert functions for special algebras A such as local complete intersections or more generally Gorenstein algebras. The sequences that occur as the Hilbert functions of standard graded complete intersections are well understood classically thanks to Macaulay and Stanley. Very little is known in the local case except in codimension two. In this paper we characterise the Hilbert functions of quadratic Artinian complete intersections of codimension three. Interestingly we prove that a Hilbert function is admissible for such a Gorenstein ring if and only if is admissible for such a complete intersection. We provide an effective construction of a local complete intersection for a given Hilbert function. We prove that the symmetric decomposition of such a complete intersection ideal is determined by its Hilbert function.     

Empty Simplices of Polytopes and Graded Betti Numbers

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope and applying a result of Migliore and the author. This approach allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.     

New results on the lower central series quotients of a free associative algebra

We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated in Feigin and Shoikhet (2007) [FS]. We establish a linear bound on the degree of tensor field modules appearing in the Jordan–Hölder series of each graded component. We also bound the leading coefficient of the Hilbert polynomial of each graded component. As applications, we confirm conjectures of P. Etingof and B. Shoikhet concerning the structure of the third graded component.     

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.     

The hilbert function and the minimal free resolution of some gorenstein ideals of codimension 4

We give a construction for Gorenstein ideals of codimension 4 which we believe have maximal graded Betti numbers for their Hilbert function.     

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.     

On the spectral analysis of many-body systems

We describe the essential spectrum and prove the Mourre estimate for quantum particle systems interacting through k-body forces and creation-annihilation processes which do not preserve the number of particles. For this we compute the “Hamiltonian algebra” of the system, i.e. the C^∗-algebra C generated by the Hamiltonians we want to study, and show that, as in the N-body case, it is graded by a semilattice. Hilbert C^∗-modules graded by semilattices are involved in the construction of C. For example, if we start with an N-body system whose Hamiltonian algebra is CN and then we add field type couplings between subsystems, then the many-body Hamiltonian algebra C is the imprimitivity algebra of a graded Hilbert CN-module.     

Remarks on Hilbert series of graded modules over polynomial rings

In this article we discuss a result on formal Laurent series and some of its implications for Hilbert series of finitely generated graded modules over standard-graded polynomial rings: For any integer Laurent function of polynomial type with non-negative values the associated formal Laurent series can be written as a sum of rational functions of the form ${\frac{Q_j(t)}{(1-t)^j}}$ , where the numerators are Laurent polynomials with non–negative integer coefficients. Hence any such series is the Hilbert series of some finitely generated graded module over a suitable polynomial ring ${\mathbb{F}[X_1 , \ldots , X_n]}$ . We give two further applications, namely an investigation of the maximal depth of a module with a given Hilbert series and a characterization of Laurent polynomials which may occur as numerator in the presentation of a Hilbert series as a rational function with a power of (1 ? t) as denominator.     

Some remarks on hilbert functions of veronese algebras

We study the Hilbert polynomials of finitely generated graded algebras R, with generators not all of degree one (i.e. non-standard). Given an expression P(R,t)=a(t)/(1-t^l ) ⁿ for the Poincare series of R as a rational function, we study for 0 ≤ i ≤ l the graded subspaces ? _kR_kl+i (which we denote R[l;i]) of R, in particular their Poincaré series and Hilbert functions. We prove, for example, that if R is Cohen-Macaulay then the Hilbert polynomials of all non-zeroR[l;i] share a common degree. Furthermore, if R is also a domain then these Hilbert polynomials have the same leading coefficient.     

Smooth and irreducible multigraded Hilbert schemes

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x,y], which establishes a conjecture of Haiman and Sturmfels.     

On the reducibility of the postulation Hilbert scheme

We give a condition in terms of the possible graded Betti numbers compatible with a given Hilbert functionH of 0-dimensional subschemes of ℙ ⁿ which implies the reducibility of the postulation Hilbert scheme and of its subscheme which parametrizes reduced subschemes with Hilbert functionH.     

Free and cofree Hopf algebras

We first prove that a graded, connected, free and cofree Hopf algebra is always self-dual. Then, we prove that two graded, connected, free and cofree Hopf algebras are isomorphic if and only if they have the same Poincaré–Hilbert formal series. If the characteristic of the base field is zero, we prove that the Lie algebra of the primitive elements of such an object is free, and we deduce a characterization of the formal series of free and cofree Hopf algebras by a condition of growth of the coefficients. We finally show that two graded, connected, free and cofree Hopf algebras are isomorphic as (nongraded) Hopf algebras if and only if the Lie algebras of their primitive elements have the same number of generators.     

Gotzmann ideals of the polynomial ring

Let A =  K[x ₁,..., x _n ] denote the polynomial ring in n variables over a field K. We will classify all the Gotzmann ideals of A with at most n generators. In addition, we will study Hilbert functions H for which all homogeneous ideals of A with the Hilbert function H have the same graded Betti numbers. These Hilbert functions will be called inflexible Hilbert functions. We introduce the notion of segmentwise critical Hilbert functions and show that segmentwise critical Hilbert functions are inflexible. The first author is supported by JSPS Research Fellowships for Young Scientists.     

Standard Bigraded Hilbert Functions

We show Macaulay-type bounds and persistence results for bigraded Hilbert function along rays. Global behavior of the bigraded Hilbert function and very supportive sets for bigraded Hilbert polynomials are also discussed.

We indicate by several results that direct generalizations of the standard graded theory are not true. In particular, we give a counterexample to an immediate generalization of Gotzmann's Regularity Theorem and Persistence Theorem. It is also made clear that maximal growth of a bigraded Hilbert function from bidegree (u, v) to bidegree (u + 1, v) usually is obtained at the expense of the growth from bidegree (u, v) to bidegree (u, v + 1).     

On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups

In [Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem [Proc. Sympos. Pure. Math. 15 (1970) 213-222], it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations.     

On the coefficients of Hilbert quasipolynomials

The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert-Serre). We derive an upper bound for its grade, i.e. the index from which on its coefficients are constant. As an application, we give a purely algebraic proof of an old combinatorial result (due to Ehrhart, McMullen and Stanley).