A Lower Bound on the Second Fiber Coefficient of the 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 and r(I | K) < ∞, we present a lower bound on the second fiber coefficient of the fiber cones, and also provide a characterization, in terms of f ₂(I, K), of the condition depth F _K (I) ≥ d ? 1.     

关于纤维锥的Hilbert系数

Let （R, m） be a Cohen-Macaulay local ring of dimension d, I an mprimary ideal and K an ideal containing I. Assuming that I has minimal （almost minimal） mixed multiplicity with respect to K, we get some results on the Hilbert coefficients of fiber cones.     

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.     

On the Multiplicity and the Cohen-Macaulayness of Fiber Cones of Graded Modules

Let (A,m) be a Noetherian local ring with maximal ideal m,I an ideal of A,J an m-primary ideal of A,N a finitely generated A-module,S =⊕n≥0 Sn a finitely generated standard graded algebra over A and M =⊕n≥0 Mn a finitely generated graded S-module.We characterize the multiplicity and the Cohen-Macaulayness of the fiber cone FJ(M) =⊕n≥0 Mn/JMn.As an application,we obtain some results on the multiplicity and the Cohen-Macaulayness of the fiber cone ⊕n≥0 InN/JInN.     

On Hilbert Coefficients of Filtrations

Let F be a Hilbert filtration with respect to a Cohen-Macaulay module M. When G(F,M)and F_K(F,M)have almost maximal depths,the Hilbert coefficients g_i(F,M)is calculated.In the general case,an upper bound for g_2(F,M)is also given.     

Primes and Right Ideals in Right Cones

A right cone H in a group G is a submonoid of G that generates G and aH ? bH for a, b ? H implies bH ? aH. With any right ideal I ≠ H of H a completely prime ideal P _r (I) of H is associated and the set 𝒫(I) of right ideals I′ of H with the same associated prime ideal P′ =P _r (I) is determined if P′·? P″ is a right invariant segment in H. The set 𝒫(I) is also described if P _r (I) is a limit prime.     

拟平移不变拓扑锥与局部β-凸空间的共轭锥

[1]中提出的局部β-凸分析问题从本质上来说是一种非线性凸分析问题 .为了刻画和研究局部β-凸空间 X的共轭锥 X*β ,本文在抽象凸锥上引进具有拟平移不变性质的拓扑结构 ,第一部分重点研究局部生成拓扑锥与赋范拓扑锥 .第二部分将这两种拓扑锥的一般理论应用于局部 β - 凸空间的共轭锥 X*β 的研究 ,得到 (X*β,U| A)与 (X*β ,‖‖ )的局部生成性与完备性定理等 .     

Depth and Stanley depth of the edge ideals of square paths and square cycles

In this paper, we compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square path on n vertices. We also compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square cycle on n vertices, when n≡0,3,4( mod 5), and give tight bounds when n≡1,2( mod 5). We also prove a conjecture of Herzog presented in [5 Herzog, J. (2013). A survey on Stanley depth. In: Bigatti, M. A., Gimenez, P., Sáenz-de-Cabezón, E., eds. Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, Vol. 2083. Heidelberg: Springer, pp. 3–45. https://arxiv.org/pdf/1702.00781.pdf.[Crossref] , [Google Scholar]], for the edge ideals of square paths and square cycles.     

A bracket power characterization of analytic spread one ideals

The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let be regular nonunits in a local (Noetherian) ring and assume that , the integral closure of , where . Then the main result shows that for all but finitely many units in that are non-congruent modulo and for all large integers and it holds that for and not divisible by , where is the -th bracket power of . And, conversely, if there exist positive integers , , and such that has a basis such that , then has analytic spread one.

     

赋范线性空间中包含方程可解性定理的推广及弱切锥的应用

本文首先在一般的赋范线性空间中研究了集值映射F;X→Y的平衡点的存在性问题,证明了包含问题O∈F(X)的三个可解性定理.然后在无穷维空间中研究了弱相依锥T_k^σ(x)的直接像,弱相依导数D^σF(x,y)的一个链式法则以及偏y弱导数D_y^σF(x,y)的弱Lip连续性.最后,作为应用,给出并证明了用弱相依导数D^σF(x,y)及弱P导数P^σF(x,y)判定无穷维     

Properties of the Fiber Cone of Ideals in Local Rings

Abstract

For an ideal I of a Noetherian local ring (R, m ) we consider properties of I and its powers as reflected in the fiber cone F(I) of I. In particular,we examine behavior of the fiber cone under homomorphic image R → R/J = R′ as related to analytic spread and generators for the kernel of the induced map on fiber cones ψ _J  : F _R (I) → F _R′(IR′). We consider the structure of fiber cones F(I) for which ker ψ _J  ≠ 0 for each nonzero ideal J of R. If dim F(I) = d > 0,μ(I) = d + 1 and there exists a minimal reduction J of I generated by a regular sequence,we prove that if grade(G ₊(I)) ≥ d ? 1,then F(I) is Cohen-Macaulay and thus a hypersurface.     

Depth and Stanley Depth of Multigraded Modules

We study the behavior of depth and Stanley depth along short exact sequences of multigraded modules and under reduction modulo an element.     

Covering morphisms

Let (R, m) be a two-dimensional regular local ring and let A be a finitely generated torsion-free R-module. If A is a complete module, then Dan Katz and Vijay Kodiyalam show A satisfies five conditions. They ask whether these five conditions are equivalent without assuming A to be complete. In a previous paper we determined all implications among these five conditions with one exception. In this paper, with an additional hypothesis on the units of R, we resolve the remaining case.     

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.     

Depths of Powers of the Edge Ideal of a Tree

Lower bounds are given for the depths of R/I ^t for t ≥ 1 when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.     

Characterization of Generating Ideals in Some Rings of Entire Functions

Let E be a ring of entire functions on with the operation of pointwise multiplication, and let f ₁,...,f _m be a set of nonzero elements in E. The ideal E(f ₁,...,f _m ) in E with generators f ₁,...,f _m is said to be generating if E(f ₁,...,f _m ) = E. The generating ideals in rings of entire functions on determined by the growth of their maximum moduli are characterized in terms of the distribution of the zero sets of their generators. Under the additional condition of rapid variation of the weight sequences determining the ring, criteria for generating ideals are established; they are stated in terms of d(z) max _{1 j m} d _j (z), where d _j (z) is the distance from a point to the zero set of f _j for 1 j m. It is shown that, in rings of entire functions of finite or minimal type with respect to a given order, a similar characterization (i.e., in terms of d(z)) cannot be given.     

$$(\vec k,\vec s)$$ -Monotone Coefficients in Spaces with Mixed Quasinorm

We estimate the sums of double series in sines and cosines with multiply monotone coefficients and other more general coefficients. We obtain necessary and sufficient conditions for membership of the sums of these series in the spaces with mixed quasinorm.