Local rings of minimal length

This paper deals with local rings R possessing an m-canonical ideal ω, R⊆ω. In particular those rings such that the length l_R(ω/R) is as short as possible are studied. The same notion for one-dimensional local Cohen-Macaulay rings was introduced and studied with the name of Almost Gorenstein. Some necessary conditions, that become also sufficient under additional hypotheses, are given and examples are provided also in the non-Noetherian case. The case when the maximal ideal of R is stable is also studied.     

Degree bounds for Gröbner bases in algebras of solvable type

We establish doubly-exponential degree bounds for Gröbner bases in certain algebras of solvable type over a field (as introduced by Kandri-Rody and Weispfenning). The class of algebras considered here includes commutative polynomial rings, Weyl algebras, and universal enveloping algebras of finite-dimensional Lie algebras. For the computation of these bounds, we adapt a method due to Dubé based on a generalization of Stanley decompositions. Our bounds yield doubly-exponential degree bounds for ideal membership and syzygies, generalizing the classical results of Hermann and Seidenberg (in the commutative case) and Grigoriev (in the case of Weyl algebras).     

Finite-sample inference with monotone incomplete multivariate normal data, II

We continue our recent work on inference with two-step, monotone incomplete data from a multivariate normal population with mean and covariance matrix . Under the assumption that is block-diagonal when partitioned according to the two-step pattern, we derive the distributions of the diagonal blocks of and of the estimated regression matrix, . We represent in terms of independent matrices; derive its exact distribution, thereby generalizing the Wishart distribution to the setting of monotone incomplete data; and obtain saddlepoint approximations for the distributions of and its partial Iwasawa coordinates. We prove the unbiasedness of a modified likelihood ratio criterion for testing , where is a given matrix, and obtain the null and non-null distributions of the test statistic. In testing , where and are given, we prove that the likelihood ratio criterion is unbiased and obtain its null and non-null distributions. For the sphericity test, , we obtain the null distribution of the likelihood ratio criterion. In testing we show that a modified locally most powerful invariant statistic has the same distribution as a Bartlett-Pillai-Nanda trace statistic in multivariate analysis of variance.     

Universal central extension and the second invariant of homology of crossed modules in lie algebras

In this paper we show that the kernel of the universal central extension of a crossed module in Lie algebras is the second invariant of this crossed module. As a consequence of this result we obtain a recognition criterion for universal central extensions and a vanishing situation of two invariants associated to a crossed module in Lie algebras.     

Int-decomposable algebras

Let D be a Dedekind domain with fraction field k. Let A be a D-algebra that, as a D-module, is free of finite rank. Let B be the extension of A to a k-algebra. The set of integer-valued polynomials over A   is defined to be Int(A)={f∈B[x]|f(A)⊆A}$\mathrm{Int}(A)=\{f\in B[x]|f(A)\subseteq A\}$. Restricting the coefficients to elements of k  , we obtain the commutative ring _Intk(A)={f∈k[x]|f(A)⊆A}${\mathrm{Int}}_k(A)=\{f\in k[x]|f(A)\subseteq A\}$; this makes Int(A)$\mathrm{Int}(A)$ a left _Intk(A)${\mathrm{Int}}_k(A)$-module. Previous researchers have noted instances when a D-module basis for A   is also an _Intk(A)${\mathrm{Int}}_k(A)$-basis for Int(A)$\mathrm{Int}(A)$. We classify all the D-algebras A   with this property. Along the way, we prove results regarding Int(A)$\mathrm{Int}(A)$, its localizations at primes of D, and finite residue rings of A.     

Epsilon multiplicity for graded algebras

The notion of ε-multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that the relative ε-multiplicity of reduced standard graded algebras over an excellent local ring exists as a limit. We also obtain some important special cases of Cutkosky's results concerning ε-multiplicity, as corollaries of our main theorem.     

Desingularization of regular algebras

We identify families of commutative rings that can be written as a direct limit of a directed system of noetherian regular rings and investigate the homological properties of such rings.     

Filament sets and homogeneous continua

New tools are introduced for the study of homogeneous continua. The subcontinua of a given continuum are classified into three types: filament, non-filament, and ample, with ample being a subcategory of non-filament. The richness of the collection of ample subcontinua of a homogeneous continuum reflects where the space lies in the gradation from being locally connected at one extreme to indecomposable at another. Applications are given to the general theory of homogeneous continua and their hyperspaces.     

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⁺.     

The multiplicity and the Cohen-Macaulayness of extended Rees algebras of equimultiple ideals

Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T=A[It,t^-1]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I) over a Cohen-Macaulay ring in terms of reduction numbers and ideals.     

Improving the bounds of the multiplicity conjecture: The codimension 3 level case

The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra A in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of A. All the examples studied so far have lead to conjecture (see [J. Herzog, X. Zheng, Notes on the multiplicity conjecture. Collect. Math. 57 (2006) 211-226] and [J. Migliore, U. Nagel, T. Römer, Extensions of the multiplicity conjecture, Trans. Amer. Math. Soc. (preprint: math.AC/0505229) (in press)]) that, moreover, the bounds of the MC are sharp if and only if A has a pure MFR. Therefore, it seems a reasonable-and useful-idea to seek better, if possibly ad hoc, bounds for particular classes of Cohen-Macaulay algebras.In this work we will only consider the codimension 3 case. In the first part we will stick to the bounds of the MC, and show that they hold for those algebras whose h-vector is that of a compressed algebra.In the second part, we will (mainly) focus on the level case: we will construct new conjectural upper and lower bounds for the multiplicity of a codimension 3 level algebra A, which can be expressed exclusively in terms of the h-vector of A, and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of A is not pure.Even though proving our bounds still appears too difficult a task in general, we are already able to show them for some interesting classes of codimension 3 level algebras A: namely, when A is compressed, or when its h-vector h(A) ends with (…,3,2). Also, we will prove our lower bound when h(A) begins with (1,3,h₂,…), where h₂≤4, and our upper bound when h(A) ends with (…,h_c−1,h_c), where h_c−1≤h_c+1.     

Derivations of polynomial algebras without Darboux polynomials

We present several new examples of homogeneous derivations of a polynomial ring k[X]=k[x₁,…,x_n] over a field k of characteristic zero without Darboux polynomials. Using a modification of a result of Shamsuddin, we produce these examples by induction on the number n of variables, thus more easily than the previously known example multidimensional Jouanolou systems of ?o?a?dek.     

On the computation of the jdeg of blowup algebras

For a graded algebra , its is a global degree that can be used to study issues of complexity of the normalization . Here some techniques grounded on Rees algebra theory are used to estimate . A closely related notion, of divisorial generation, is introduced to count numbers of generators of .     

Buchsbaum Stanley-Reisner rings with minimal multiplicity

In this paper, we study Buchsbaum Stanley-Reisner rings with linear free resolution. We introduce the notion of Buchsbaum Stanley-Reisner rings with minimal multiplicity of initial degree , which extends the notion of Buchsbaum rings with minimal multiplicity defined by Goto. As an application, we give many examples of non-Cohen-Macaulay Buchsbaum Stanley-Reisner rings with linear resolution.

     

Über Schwache Sequenzen

Ohne Zusammenfassung

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Complexity of the normalization of algebras

We introduce and develop new techniques to study the complexity of normalization processes of graded algebras. The construction of a new degree function on graded modules, with a global nature, permits a broad extension of recent bounds for the length of the chains of subalgebras that general algorithms must transverse to build the integral closure, particularly of blowup algebras. It achieves this by relating the values of the new degree with invariants of the algebra known ab initio. As a by-product, it reveals new inequalities among Hilbert coefficients. The second author was partially supported by the NSF.     

Mixed multiplicities and the minimal number of generator of modules

Let (R,m) be a d-dimensional Noetherian local ring. In this work we prove that the mixed Buchsbaum-Rim multiplicity for a finite family of R-submodules of R^p of finite colength coincides with the Buchsbaum-Rim multiplicity of the module generated by a suitable superficial sequence, that is, we generalize for modules the well-known Risler-Teissier theorem. As a consequence, we give a new proof of a generalization for modules of the fundamental Rees’ mixed multiplicity theorem, which was first proved by Kirby and Rees in (1994, [8]). We use the above result to give an upper bound for the minimal number of generators of a finite colength R-submodule of R^p in terms of mixed multiplicities for modules, which generalize a similar bound obtained by Cruz and Verma in (2000, [5]) for m-primary ideals.     

Generic initial ideals and graded Artinian-level algebras not having the Weak-Lefschetz Property

We find a sufficient condition that is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function cannot be level if h_d≤2d+3, and that there exists a level O-sequence of codimension 3 of type for h_d≥2d+k for k≥4. Furthermore, we show that is not level if , and also prove that any codimension 3 Artinian graded algebra A=R/I cannot be level if . In this case, the Hilbert function of A does not have to satisfy the condition h_d−1>h_d=h_d+1.Moreover, we show that every codimension n graded Artinian level algebra having the Weak-Lefschetz Property has a strictly unimodal Hilbert function having a growth condition on (h_d−1−h_d)≤(n−1)(h_d−h_d+1) for every d>θ where

h₀<h₁<<h_α==h_θ>>h_s−1>h_s.