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.     

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.     

Hilbert coefficients and Buchsbaumness of associated graded rings

Let A be a Noetherian local ring with the maximal ideal and I an -primary ideal. The purpose of this paper is to generalize Northcott's inequality on Hilbert coefficients of I given in Northcott (J. London Math. Soc. 35 (1960) 209), without assuming that A is a Cohen-Macaulay ring. We will investigate when our inequality turns into an equality. It is related to the Buchsbaumness of the associated graded ring of I.     

Factoring ideals in Prüfer domains

We show that in certain Prüfer domains, each nonzero ideal I can be factored as , where I^v is the divisorial closure of I and is a product of maximal ideals. This is always possible when the Prüfer domain is h-local, and in this case such factorizations have certain uniqueness properties. This leads to new characterizations of the h-local property in Prüfer domains. We also explore consequences of these factorizations and give illustrative examples.     

Quasi-socle ideals in a Gorenstein local ring

This paper explores the structure of quasi-socle ideals I=Q:m² in a Gorenstein local ring A, where Q is a parameter ideal and m is the maximal ideal in A. The purpose is to answer the problems as to when Q is a reduction of I and when the associated graded ring is Cohen-Macaulay. Wild examples are explored.     

Quasi-socle ideals and Goto numbers of parameters

Goto numbers for certain parameter ideals Q in a Noetherian local ring (A,m) with Gorenstein associated graded ring are explored. As an application, the structure of quasi-socle ideals I=Q:m^q (q≥1) in a one-dimensional local complete intersection and the question of when the graded rings are Cohen-Macaulay are studied in the case where the ideals I are integral over Q.     

A uniform Artin-Rees property for syzygies in rings of dimension one and two

Let be a local Noetherian ring, let M be a finitely generated R-module and let I⊂R be an -primary ideal. Let be a free resolution of M. In this paper we study the question whether there exists an integer h such that IⁿF_i∩ker(∂_i)⊂I^n−hker(∂_i) holds for all i. We give a positive answer for rings of dimension at most two. We relate this property to the existence of an integer s such that I^s annihilates the modules for all i>0 and all integers n.     

Strong lifting splits

The concept of an enabling ideal is introduced so that an ideal I is strongly lifting if and only if it is lifting and enabling. These ideals are studied and their properties are described. It is shown that a left duo ring is an exchange ring if and only if every ideal is enabling, that Zhou’s δ-ideal is always enabling, and that the right singular ideal is enabling if and only if it is contained in the Jacobson radical. The notion of a weakly enabling left ideal is defined, and it is shown that a ring is an exchange ring if and only if every left ideal is weakly enabling. Two related conditions, interesting in themselves, are investigated: the first gives a new characterization of δ-small left ideals, and the second characterizes weakly enabling left ideals. As an application (which motivated the paper), let M be an I-semiregular left module where I is an enabling ideal. It is shown that m∈M is I-semiregular if and only if m−q∈IM for some regular element q of M and, as a consequence, that if M is countably generated and IM is δ-small in M, then where for each i.     

Stable cohomology over local rings

For a commutative noetherian ring R with residue field k stable cohomology modules have been defined for each n∈Z, but their meaning has remained elusive. It is proved that the k-rank of any characterizes important properties of R, such as being regular, complete intersection, or Gorenstein. These numerical characterizations are based on results concerning the structure of Z-graded k-algebra carried by stable cohomology. It is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras . Some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings.     

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].     

On the diameter and girth of a zero-divisor graph

For a commutative ring R with zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we characterize when either or . We then use these results to investigate the diameter and girth for the zero-divisor graphs of polynomial rings, power series rings, and idealizations.     

Homology of artinian and Matlis reflexive modules, I

Let R be a commutative local noetherian ring, and let L and L^′ be R-modules. We investigate the properties of the functors and . For instance, we show the following:

(a)

if L and L^′ are artinian, then is artinian, and is noetherian over the completion ;

(b)

if L is artinian and L^′ is Matlis reflexive, then , , and are Matlis reflexive.

Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.     

Localization and duality in topology and modular representation theory

We develop a duality theory for localizations in the context of ring spectra in algebraic topology. We apply this to prove a theorem in the modular representation theory of finite groups.Let G be a finite group and k be an algebraically closed field of characteristic p. If p is a homogeneous nonmaximal prime ideal in H^∗(G,k), then there is an idempotent module κ_p which picks out the layer of the stable module category corresponding to p, and which was used by Benson, Carlson and Rickard [D.J. Benson, J.F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997) 59-80] in their development of varieties for infinitely generated kG-modules. Our main theorem states that the Tate cohomology is a shift of the injective hull of H^∗(G,k)/p as a graded H^∗(G,k)-module. Since κ_p can be constructed using a version of the stable Koszul complex, this can be viewed as a statement of localized Gorenstein duality in modular representation theory. Various consequences of this theorem are given, including the statement that the stable endomorphism ring of the module κ_p is the p-completion of cohomology , and the statement that κ_p is a pure injective kG-module.In the course of proving the theorem, we further develop the framework introduced by Dwyer, Greenlees and Iyengar [W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357-402] for translating between the unbounded derived categories and . We also construct a functor to the full stable module category, which extends the usual functor and which preserves Tate cohomology. The main theorem is formulated and proved in , and then translated to and finally to .The main theorem in can be viewed as stating that a version of Gorenstein duality holds after localizing at a prime ideal in H^∗(BG;k). This version of the theorem holds more generally for a compact Lie group satisfying a mild orientation condition. This duality lies behind the local cohomology spectral sequence of Greenlees and Lyubeznik for localizations of H^∗(BG;k).In a companion paper [D.J. Benson, Idempotent kG-modules with injective cohomology, J. Pure Appl. Algebra 212 (7) (2008) 1744-1746], a more recent and shorter proof of the main theorem is given. The more recent proof seems less natural, and does not say anything about localization of the Gorenstein condition for compact Lie groups.     

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.     

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.     

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.     

Quadratic addition rules for quantum integers

For every positive integer n, the quantum integer [n]_q is the polynomial [n]_q=1+q+q²+?+q^n-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation .     

Syzygies and the Rees algebra

Let a,b,c be linearly independent homogeneous polynomials in the standard Z-graded ring R?k[s,t] with the same degree d and no common divisors. This defines a morphism P¹→P². The Rees algebra of the ideal I=〈a,b,c〉 is the graded R-algebra which can be described as the image of an R-algebra homomorphism h: . This paper discusses one result concerning the structure of the kernel of the map h and its relation to the problem of finding the implicit equation of the image of the map given by a, b, c. In particular, we prove a conjecture of Hong, Simis and Vasconcelos. We also relate our results to the theory of adjoint curves and prove a special case of a conjecture of Cox.     

Images of locally finite derivations of polynomial algebras in two variables

In this paper we show that the image of any locally finite k-derivation of the polynomial algebra k[x,y] in two variables over a field k of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image of every k-derivation D of k[x,y] such that and is a Mathieu subspace of k[x,y].