Stickelberger ideals and divisor class numbers

Let K/k be a finite abelian extension of function fields with Galois group G. Using the Stickelberger elements associated to K/k studied by J. Tate, P. Deligne and D. Hayes, we construct an ideal I in the integral group ring relative to the extension K/k, whose elements annihilate the group of divisor classes of degree zero of K and whose rank is equal to the degree of the extension. When K/k is a (wide or narrow) ray class extension, we compute the index of I in , which is equal to the divisor class number of K up to a trivial factor. Received: 4 November 1999; in final form: 8 September 2000 / Published online: 23 July 2001     

Stickelberger ideals and Fitting ideals of class groups for abelian number fields

In this paper, we determine completely the initial Fitting ideal of the minus part of the ideal class group of an abelian number field over Q up to the 2-component. This answers an open question of Mazur and Wiles (Invent Math 76:179–330, 1984) up to the 2-component, and proves Conjecture 0.1 in Kurihara (J Reine Angew Math 561:39–86, 2003). We also study Brumer’s conjecture and prove a stronger version for a CM-field, assuming certain conditions, in particular on the Galois group.     

The indices of Stickelberger ideals of function fields

Recently Yin [Y4] calculated the index of Stickelberger ideal in the group ring when is a (wide or narrow) ray class extension under the assumption that . In this paper, we extend Yin's result to general Received: 23 August 2001 / Published online: 27 June 2002     

Stickelberger elements,Fitting ideals of class groups of CM-fields,and dualisation

In this paper, we systematically construct abelian extensions of CM-fields over a totally real field whose Stickelberger elements are not in the Fitting ideals of the class groups. Our evidence indicates that Pontryagin duals of class groups behave better than the class groups themselves. We also explore the behaviour of Fitting ideals under projective limits and dualisation in a somewhat broader context.     

Cyclotomic units and Stickelberger ideals of global function fields

In this paper, we define the group of cyclotomic units and Stickelberger ideals in any subfield of the cyclotomic function field. We also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of Stickelberger ideals in the integral group ring.

     

On Whitney numbers of the order ideals of generalized fences and crowns

We solve some recurrences given by E. Munarini and N. Zagaglia Salvi proving explicit formulas for Whitney numbers of the distributive lattices of order ideals of the fence poset and crown poset. Moreover, we get a method to obtain explicit formulas for Whitney numbers of lattices of order ideals of fences with higher asymmetric peaks.     

Reduction numbers and initial ideals

The reduction number of a standard graded algebra is the least integer such that there exists a minimal reduction of the homogeneous maximal ideal of such that . Vasconcelos conjectured that where is the initial ideal of an ideal in a polynomial ring with respect to a term order. The goal of this note is to prove the conjecture.

     

Betti numbers of determinantal ideals

Let R=k[x₁,…,x_n] be a polynomial ring and let IR be a graded ideal. In [T. Römer, Betti numbers and shifts in minimal graded free resolutions, arXiv: AC/070119], Römer asked whether under the Cohen–Macaulay assumption the ith Betti number β_i(R/I) can be bounded above by a function of the maximal shifts in the minimal graded free R-resolution of R/I as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen–Macaulay algebras k[x₁,…,x_n]/I when I is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds are sharp.     

On Eisenstein ideals and the cuspidal group of J0(N)

Let C_N be the cuspidal subgroup of the Jacobian J₀(N) for a square-free integer N > 6. For any Eisenstein maximal ideal m of the Hecke ring of level N, we show that C_N[m] ≠ 0. To prove this, we calculate the index of an Eisenstein ideal I contained in m by computing the order of the cuspidal divisor annihilated by I.     

广义Bernoulli数和广义高阶Bernoulli数

定义了广义Bernoulli数和广义高阶Bernoulli数，建立了它们的递推公式和有关性质，从而推广了Bernoulli数和高阶Bernoulli数。     

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.     

Intersection numbers of families of ideals

We study the intersection number of families of tall ideals. We show that the intersection number of the class of analytic P-ideals is equal to the bounding number ${\mathfrak{b}}$ , the intersection number of the class of all meager ideals is equal to ${\mathfrak{h}}$ and the intersection number of the class of all F _σ ideals is between ${\mathfrak{h}}$ and ${\mathfrak{b}}$ , consistently different from both.     

Betti numbers of mixed product ideals

We compute the Betti numbers of the resolution of a special class of square-free monomial ideals, the ideals of mixed products. Moreover when these ideals are Cohen-Macaulay we calculate their type. Received: 9 March 2008     

Extremal Betti numbers of graded ideals

The possible extremal Betti numbers of graded ideals in the polynomial ring K[x₁,…,x_n] in n variables with coefficients in a field K are studied, completing our results in [7]. In case char(K) = 0 we determine, given any integers r < n, the conditions under which there exists a graded ideal I ? K[x₁,…, x_n] with extremal Betti numbers $\beta_{k_{i}k_{i}+\ell_{i}}\ {\rm for}\ i=1,\cdots,r$ . We also treat a similar problem for squarefree lexsegment ideals.     

Extremal Betti numbers of edge ideals

Archiv der Mathematik - Let A be an algebra over a field F of characteristic zero. For every $$n\ge 1$$ , let $$\delta _n(A)$$ be the number of linearly independent multilinear proper central...     

Betti numbers of some monomial ideals

In this paper we compute the graded Betti numbers of certain monomial ideals that are not stable. As a consequence we prove a conjecture, stated by G. Fatabbi, on the graded Betti numbers of two general fat points in

     

Betti numbers of perfect homogeneous ideals

In this paper we study the graded minimal free resolution of the ideal, I, of any arithmetically Cohen-Macaulay projective variety. First we determine the range of the shifts (twisting numbers) that can possibly occur in the resolution, in terms of the Hilbert function of I. Then we find conditions under which some of the twisting numbers do not occur. Finally, in some ‘good’ cases, all the Betti numbers are (recursively) computed, in terms of the Hilbert function of I or that of Extⁿ_R(R/I,R), where R is a polynomial ring over a field and n is the height of I in R.     

High order derivations and primary ideals to regular prime ideals

Ohne Zusammenfassung     

On a class of rational cuspidal plane curves

We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.