The conductor of one-dimensional gorenstein rings in their blowing-up

Let (A,M) be a local, one-dimensional, Cohen-Macaulay ring of multiplicity e=e(A)>1 and Hilbert function H(A). Let I=Ann_A (B/A) be the conductor of A in its blowing up B. Northcott and Matlis have proved that if the embedding dimension emdim A of A is 2 then I=M^e−1 [3; Corollary 13.8]. If emdim A>2 little is know about I. In [6] and [7] I is computed when the associated graded ring G(A) is reduced (in this case B in the integral closure of A). In this paper we compute I when A is Gorenstein. There are in general upper and lower bounds for I in terms of a power of M and we start discussing when these bounds are attained. In particular we show that in the extremal situation I=M^e−1 one has emdimA=2 (thus inverting the result of Northcott and Matlis). Then we consider the case of Gorenstein rings. We prove that if G(A) in Gorenstein then I=M^ϑ where ϑ=Min{n‖H(n)=e}. If more generally A is Gorenstein then I⊂M² or emdim A=e(A)=2. When A is the local ring of a curve at a singular point p we get, as a consequence of this last result a proof of the following conjecture of Catanese which has interesting geometric applications [1]: if the conductor J of A in its normalization is not contained in M² then p is a node.     

A SURJECTIVE HOMOMORPHISM OF LOCAL COHOMOLOGY MODULES

Let R be a local ring and M a finitely generated generalized Cohen-Macaulay R-module such that dim _R M = dim _R M/αM + height_Mα a for all ideals α of R. Suppose that H_I ^j(M) ≠ 0 for an ideal I of R and an integer j > height_M I. We show that there exists an ideal J ? such that a. height_M J = j;

b. the natural homomorphism H_I ^j(M) → H_I ^j(M) is an isomorphism, for all i > j; and,

c. the natural homomorphism H_I ^j(M) → H_I ^j(M) is surjective.

By using this theorem, we obtain some results about Betti numbers, coassociated primes, and support of local cohomology modules.     

Rees algebras with minimal multiplicity

Let (R,m) be a local ring. Let S_M denote the Rees algebra S=R[m^rt] localized at its unique maximal homogeneous ideal M=(m,m^rt). Let TN denote the extended Rees algebra T= R[m^rt, t^-1] localized at its unique maximal homogeneous idea N= (t^?1,m,m^r). Multiplicity formulas are developedfor S_M and T_N. These are used to find necessaIy and sufficient conditions on a Cohen-Macaulay local ring (R,m) and r so that S_M and T_N are Cohen-Macaulay with minimal multiplicity     

Typical sheaves of generalized CM-modules

Let M be a generalized Cohen-Macaulay module over a noetherian local ring (R,m). Fix a standard system x₁, …, x_d∈m with respect to M and let . We construct a coherent Cohen-Macaulay sheafK over the projective space ℙ _R/I ^d-1 whose cohomological Hilbert functions depend only on the lengths of the local cohomology modules H _m ⁱ (M), (i=0, …, d−1).     

Probability measures on [SIN] groups and some related ideals in group algebras

Given a locally compact group G, let J(G){\cal J}(G) denote the set of closed left ideals in L ¹(G), of the form J _μ = [L¹(G) * (δ _e − μ)]⁻, where μ is a probability measure on G. Let J_d(G)={\cal J}_d(G)= {J_m;m is discrete}\{J_{\mu};\mu\ {\rm is discrete}\} , J_a(G)={J_m;m is absolutely continuous}{\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\} . When G is a second countable [SIN] group, we prove that J(G)=J_d(G){\cal J}(G)={\cal J}_d(G) and that J_a(G){\cal J}_a(G) , being a proper subset of J(G){\cal J}(G) when G is nondiscrete, contains every maximal element of J(G){\cal J}(G) . Some results concerning the ideals J _μ in general locally compact second countable groups are also obtained.     

Cohen-macaulay bipartite graphs

Let G be a graph on the vertex set V={x ₁, ..., x _n}. Let k be a field and let R be the polynomial ring k[x ₁, ..., x _n]. The graph ideal I(G), associated to G, is the ideal of R generated by the set of square-free monomials x _ix_j so that x _i, is adjacent to x _j. The graph G is Cohen-Macaulay over k if R/I(G) is a Cohen-Macaulay ring. Let G be a Cohen-Macaulay bipartite graph. The main result of this paper shows that G{v} is Cohen-Macaulay for some vertex v in G. Then as a consequence it is shown that the Reisner-Stanley simplicial complex of I(G) is shellable. An example of N. Terai is presented showing these results fail for Cohen-Macaulay non bipartite graphs. Partially supported by COFAA-IPN, CONACyT and SNI, México.     

Commutativity of Rings with Constraints Involving a Subset

Suppose that R is an associative ring with identity 1, J(R) the Jacobson radical of R, and N(R) the set of nilpotent elements of R. Let m 1 be a fixed positive integer and R an m-torsion-free ring with identity 1. The main result of the present paper asserts that R is commutative if R satisfies both the conditions(i) [x ^m, y ^m] = 0 for all and(ii) [(xy) ^m + y ^m x ^m, x] = 0 = [(yx) ^m + x ^m y ^m, x], for all This result is also valid if (i) and (ii) are replaced by (i) [x ^m, y ^m] = 0 for all and (ii) [(xy) ^m + y ^m x ^m, x] = 0 = [(yx) ^m + x ^m y ^m, x] for all Other similar commutativity theorems are also discussed.     

泛剩余交

本文建立了泛剩余交理论，并揭示了它与剩余交和一般剩余交的关系，得到：在交换局部环Ｒ的扩张Ｓ＝Ｒ（Ｘ）＝Ｒ［Ｘ］_{mＲ［Ｘ］}中，存在ＩＳ的一个ｓ－剩余交ＵＲＩ（ｓ；Ｉ），使得对Ｉ在Ｒ中的任意ｓ－剩余交Ｊ，ＵＲＩ（ｓ；Ｉ）是Ｊ的本质形变，且是Ｉ的一般ｓ－剩余交ＲＩ（ｓ；Ｉ）局部化ＲＩ（ｓ；Ｉ）_{mＲ［Ｘ］}．并给出了一些应用，为研究剩余交和一般剩余交提供了工具．     

Cohen-Macaulayness of Special Fiber Rings

Abstract

Let (R, 𝔪) be a Noetherian local ring and let Ibe an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ? = ?/𝔪? of I, where ? denotes the Rees algebra of I. Our key idea is to require ‘good’ intersection properties as well as ‘few’ homogeneous generating relations in low degrees. In particular, if Iis a strongly Cohen-Macaulay R-ideal with G _?and the expected reduction number, we conclude that ? is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of ?/K? for any 𝔪-primary ideal K. This result recovers a well-known criterion of Valabrega and Valla whenever K = I. Furthermore, we study the relationship between the Cohen-Macaulay property of the special fiber ring ? and the Cohen-Macaulay property of the Rees algebra ? and the associated graded ring 𝒢 of I. Finally, we focus on the integral closedness of 𝔪I. The latter question is motivated by the theory of evolutions.     

The Asymptotic Worst-Case Behavior of the FFD Heuristic for Small Items

The First-Fit-Decreasing (FFD) algorithm is one of the most famous and most studied methods for an approximative solution of the bin-packing problem. The question on the parametric behavior of the FFD heuristic for small items was raised in D. S. Johnson's thesis (1973, MIT, Cambridge, MA) and in E. G. Coffman et al. (1987, SIAM J. Comput.7, 1–17): what is the asymptotic worst-case ratio for FFD when restricted to lists with item sizes in the interval (0, α] for α ≤ . Let R^∞_FFD(α) denote the asymptotic worst-case ratio for these lists. In his thesis, Johnson gave the values of R^∞_FFD(α) for and he conjectured that

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for all integers m ≥ 4. J. Csirik (1993, J. Algorithms15, 1–28) proved that, for all integers m ≥ 5, this conjecture is true when m is even. When m is odd, he further showed where G_m ≡ 1 + (m² + m − 1)/(m(m + 1)(m + 2)) = F_m + 1/(m(m + 1)(m + 2)). These results leave open the values of R^∞_FFD(α) for 0 < α < 1/5 that are not the reciprocals of integers. In this paper we resolve the remaining open cases.     

Symbolic powers,Serre conditions and Cohen-Macaulay Rees algebras

Summary LetR be a Cohen-Macaulay ring andI an unmixed ideal of heightg which is generically a complete intersection and satisfiesI ⁽ⁿ⁾=Iⁿ for alln≥1. Under what conditions will the Rees algebra be Cohen-Macaulay or have good depth? A series of partial answers to this question is given, relating the Serre condition (S _r ) of the associated graded ring to the depth of the Rees algebra. A useful device in arguments of this nature is the canonical module of the Rees algebra. By making use of the technique of the fundamental divisor, it is shown that the canonical module has the expected form: ω _R[It] ≅(t(1−t) ^g−2). The third author was partially supported by the NSF This article was processed by the author using theLaTex style filecljour1 from Springer-Verlag.     

The unitary connections on the complex Grassmann manifold

In the complex Grassmann manifold ℱ(m,n), the space of complexn-planes passes through the origin of C^m+n; the local coordinate of the space can be arranged into anm ×n matrixZ. It is proved that

Recognition by prime graph of $^2 D_{2^m + 1} (3)$

As shown in [1] the simple group ² D_{2^m + 1} (3)^2 D_{2^m + 1} (3) is recognizable by spectrum. The main result of this paper generalizes the above, stating that ² D_{2^m + 1} (3)^2 D_{2^m + 1} (3) is recognizable by prime graph. In other words, we show that if G is a finite group satisfying G(G) = G(² D_{2^m + 1} (3))\Gamma (G) = \Gamma (^2 D_{2^m + 1} (3)) then G @ ² D_{2^m + 1} (3)G \cong ^2 D_{2^m + 1} (3).     

A general theory of structure spaces with applications to spaces of prime ideals

A structure space is a quadrupleX=(X, d, A, P), where for some setR, X A=2 ^R ,d:X×X A is defined byd(I, J)=J–I, andP is the family of cofinite subsets ofR. Forr P, I X, N _r (I)={J X: d(I, J) r},To(X)={Q X: if x Q there is anr P such thatN _r (x) Q}. ThenTo(X) is a (not usually Hausdorff) topology onX called the hull-kernel topology. Replacing d byd ^*, whered ^* (I, J)=d(J, I), or byd ^s, whered ^s (I, J.)=d(I, J) d ^* (I, J), and proceeding in the obvious way yields thedual hull-kernel topology To(X ^*) andsymmetric topology To(X ^s ). The latter is always a zero-dimensional Hausdorff space. When R is a commutative ring with identity andX is a collection of proper prime ideals ofR, To(X ^s ) is usually called thepatch topology. Our generality enables us to improve on known results in the case of space of prime ideals and to apply this theory to a wide variety of algebraic structures. In particular, we establish criteria for a subspace of a structure space to be closed in the symmetric topology; we establish a duality between families of maximal elements in the hull-kernel topology and families of minimal elements in the dual hull-kernel topology of subspaces that are closed in the symmetric topology; we use topological constructions to generalize certain ring theoretic notions, such as radical ideals an annihilator ideals; we use this theory to obtain new results about subspaces of the space prime ideals of a reduced, commutative ring.Presented by F. E. J. Linton.This author's research was supported by a grant from the CUNY-PSC research award program.     

On Fiber Coefficients of Fiber Cones †

Let (R,𝔪) be a Cohen–Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R, and K an ideal containing I. When depth G(I) ≥ d ? 1, depth F_K(I) ≥ d ? 2, and r(I|K) < ∞, we calculate the fiber coefficients f_i(I). Under the above assumptions on depth G(I) and r(I|K), we give an upper bound for f₁(I), and also provide a characterization, in terms of f₁(I), of the condition depth F_K(I) ≥ d ? 2.     

关于纤维锥的深度和Hilbert级数

Let (R,m) be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩ J = JKIn-1 for n ≤ k ? 1 and λ( JKKIIkk-1 ) = 1. We show that if depth G(I) ≥ d-2, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of FK(I) assuming that depth G(I) ≥ d - 1.     

Parallel Multiplication in GF(2k) using Polynomial Residue Arithmetic

We present a novel method of parallelization of the multiplication operation in GF(2^k) for an arbitrary value of k and arbitrary irreducible polynomial n(x) generating the field. The parallel algorithm is based on polynomial residue arithmetic, and requires that we find L pairwise relatively prime modulim _i(x) such that the degree of the product polynomialM(x)=m ₁(x)m ₂(x)··· m_L(x) is at least 2k. The parallel algorithm receives the residue representations of the input operands (elements of the field) and produces the result in its residue form, however, it is guaranteed that the degree of this polynomial is less than k and it is properly reduced by the generating polynomial n(x), i.e., it is an element of the field. In order to perform the reductions, we also describe a new table lookup based polynomial reduction method.     

On the Cohen-Macaulay property of diagonal subalgebras of the Rees algebra

We consider the blowing up of ℙ _k ^/n−1 along a closed subscheme defined by a homogeneous idealI ∪A=k[X ₁, …,X _n ] generated by forms of degree ≤d, and its projective embeddings by the linear systems corresponding to (I ^e ) _c , forc≥de+1. The homogeneous coordinate rings of these embeddings arek[(I ^e ) _c ]. One wants to study the Cohen-Macaulay property of these rings. We will prove that if the Rees algebraR _A (I) is Cohen-Macaulay, thenk[(I ^e ) _c ] are Cohen-Macaulay forc>>e>0, thus proving a conjecture stated by A. Conca, J. Herzog, N.V. Trung and G. Valla. Supported by a F.P.I. grant of Ministerio de Educación y Ciencia (Spain) This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

On Einstein Riemannian <Emphasis Type="Italic">g</Emphasis>-natural metrics on unit tangent sphere bundles

We prove that there is always a locally homogeneous Einstein g-natural metric on the unit tangent sphere bundle over any Riemannian space of constant positive sectional curvature. Furthermore, using the (1–1) correspondence between all SO(m + 1)-invariant homogeneous metrics on the Stiefel manifold V₂ \mathbbR^m+1 = SO(m+1)/SO(m-1){V_2 \mathbb{R}^{m+1} = {{SO}}(m+1)/{{SO}}(m-1)} and all g-natural metrics on T₁ S^m{T_1 S^m} (Abbassi and Kowalski, Diff. Geom. Appl., to appear [7]), we reconstruct, by purely local procedure, the same well-known unique SO(m + 1)-invariant homogeneous Einstein metric on V₂ \mathbbR^m+1, m 1 3{V_2 \mathbb{R}^{m+1}, m \neq 3}, initially constructed by Kobayashi.