Fitting ideals of p-ramified Iwasawa modules over totally real fields

We completely calculate the Fitting ideal of the classical p-ramified Iwasawa module for any abelian extension K/k of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former results where we had to assume that only p-adic places may ramify in K/k. One of the important ingredients is the computation of some complexes in appropriate derived categories.

On the relation between torsion submodule and Fitting ideals

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a submodule of [Formula: see text] which consists of columns of a matrix [Formula: see text] with [Formula: see text] for all [Formula: see text], [Formula: see text], where [Formula: see text] is an index set. For every [Formula: see text], let I[Formula: see text] be the ideal generated by subdeterminants of size [Formula: see text] of the matrix [Formula: see text]. Let [Formula: see text]. In this paper, we obtain a constructive description of [Formula: see text] and we show that when [Formula: see text] is a local ring, [Formula: see text] is free of rank [Formula: see text] if and only if I[Formula: see text] is a principal regular ideal, for some [Formula: see text]. This improves a lemma of Lipman which asserts that, if [Formula: see text] is the [Formula: see text]th Fitting ideal of [Formula: see text] then [Formula: see text] is a regular principal ideal if and only if [Formula: see text] is finitely generated free and [Formula: see text] is free of rank [Formula: see text]

Inline Raman Spectroscopy and Indirect Hard Modeling for Concentration Monitoring of Dissociated Acid Species

We propose an approach for monitoring the concentration of dissociated carboxylic acid species in dilute aqueous solution. The dissociated acid species are quantified employing inline Raman spectroscopy in combination with indirect hard modeling (IHM) and multivariate curve resolution (MCR). We introduce two different titration-based hard model (HM) calibration procedures for a single mono- or polyprotic acid in water with well-known (method A) or unknown (method B) acid dissociation constants p Ka. In both methods, spectra of only one acid species in water are prepared for each acid species. These spectra are used for the construction of HMs. For method A, the HMs are calibrated with calculated ideal dissociation equilibria. For method B, we estimate p Ka values by fitting ideal acid dissociation equilibria to acid peak areas that are obtained from a spectral HM. The HM in turn is constructed on the basis of MCR data. Thus, method B on the basis of IHM is independent of a priori known p K a values, but instead provides them as part of the calibration procedure. As a detailed example, we analyze itaconic acid in aqueous solution. For all acid species and water, we obtain low HM errors of < 2.87 × 10−4mol mol−1 in the cases of both methods A and B. With only four calibration samples, IHM yields more accurate results than partial least squares regression. Furthermore, we apply our approach to formic, acetic, and citric acid in water, thereby verifying its generalizability as a process analytical technology for quantitative monitoring of processes containing carboxylic acids.

On Fitting Ideals of Kahler Differential Module

Let k be an algebraically closed field of characteristic zero, and R / I and S / J be algebras over k . Ω 1 ( R / I ) and Ω 1 ( S / J ) denote universal module of first order derivation over k . The main result of this paper asserts that the first nonzero Fitting ideal Ω 1 ( R / I ⊗ k S / J ) is an invertible ideal, if the first nonzero Fitting ideals Ω 1 ( R / I ) and Ω 1 ( S / J ) are invertible ideals. Then using this result, we conclude that the projective dimension of Ω 1 ( R / I ⊗ k S / J ) is less than or equal to one.

THE FRACTIONAL GALOIS IDEAL FOR ARBITRARY ORDER OF VANISHING

We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those L-functions of the extension which are non-zero at the special point s = 0, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include L-functions with a simple zero at s = 0, and was shown by the present author to provide class-group annihilators not existing in the Stickelberger ideal. The version presented in this paper deals with L-functions of arbitrary order of vanishing at s = 0, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators. Finally, we prove an equality involving Stark elements and class-groups originally due to Büyükboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof.

VALUES AT s = -1 OF L-FUNCTIONS FOR MULTI-QUADRATIC EXTENSIONS OF NUMBER FIELDS, AND THE FITTING IDEAL OF THE TAME KERNEL

Fix a Galois extension [Formula: see text] of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in [Formula: see text], let [Formula: see text] denote the primes of [Formula: see text] lying above those in S, and let [Formula: see text] denote the ring of [Formula: see text]-integers of [Formula: see text]. We then compare the Fitting ideal of [Formula: see text] as a ℤ[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of ℚ[G], and hence in ℤ[1/2][G]. Results in ℤ[G] are obtained under the assumption of the Birch–Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where [Formula: see text] is a biquadratic extension of F containing the first layer of the cyclotomic ℤ2-extension of F, and describe a class of biquadratic extensions of F = ℚ that satisfy this condition.