Balwant Singh

Generators for the derivation modules and the relation ideals of certain curves

LetO be a curve in the affine algebroide-space over a fieldK of characteristic zero. LetD be the module ofK-derivations andP the relation ideal ofO. Generators forD andP are computed in several cases. It is shown in particular that in the case of a monomial curve defined by a sequence ofe positive integers somee−1 of which form an arithmetic sequence, μO ≤ 2e - 3 and μ(P)≤e(e−1)/2.

On weak subintegrality

Abstract In a previous paper the last two authors introduced a condition which gave an elementwise characterization of subintegrality for an extension A ⊆ B of commutative Q-algebras. In the present paper we show that the same condition gives an elementwise characterization of weak subintegrality for an extension A ⊆ B of arbitrary commutative rings. We also give a new characterization of weakly subintegral elements in which the “coefficients” lie in A rather than B.

Maximally differential prime ideals in a complete local ring

Let A be a Noetherian local ring containing a field of characteristic zero. It was proved recently by Seibt (51 that if A is complete and P is a maximally differential prime ideal of A, then P is permissible in .4 (see Section 1 for terminology). In fact, Seibt has claimed this result for a more general class of rings but, as we point out in Remark 2.9, the proof given by him is valid only if A is complete or dim A/P < 1. Seibt has also given 15, Example 1.31 to show that the converse of the above result does not hold. Although this example is incorrect (because the prime ideal considered in the example is not permissible)? it is true that the converse does not hold. This is seen easily from our description of maximally differential prime ideals given in Theorem 2.6; however. see [4] for a specific example. The main result of this paper is

Hilbert Coefficients and Depths of Form Rings

Abstract We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a Cohen–Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e 1. As an application, we derive bounds on the higher Hilbert coefficient e i in terms of e 0.

Finiteness of Subintegrality

Let A and B be commutative algebras containing the rationals, with A contained in B,and B subintegral over A. In an earlier paper the authors showed that if A is excellent of finite Krull dimension then there is a natural isomorphism from B/A to the group of invertible A-submodules of B. In the present paper we remove the requirement that A be excellent of finite Krull dimension.

Differential operators on a hypersurface

On etudie des operateurs differentiels sur une variete algebrique affine dans le cadre de la conjecture de Nakai. On travaille sur un corps k de caracteristique O

Nanoporous gold–copper oxide based all-solid-state micro-supercapacitors

The rapid growth of miniaturized electronic devices has increased the demand for energy storage devices with small dimensions. Micro-supercapacitors have great potential to supplement or replace batteries and electrolytic capacitors for a wide range of applications. Micro-supercapacitor can be fabricated with micro-electronic devices for efficient energy storage unit. However, the lower energy densities of micro-supercapacitors are still a bigger challenge to its application in micro devices. In this paper, we report all-solid-state nanoporous gold (NPG)–copper oxide (CuO) based micro-supercapacitor prepared using a simple fabrication process. In this process, first NPG interdigital patterns were developed by using a simple annealing and dealloying procedure, and then CuO was electrodeposited on NPG interdigital microelectrodes. The nanoporous gold substrate provides good electronic/ionic conductivity with high intrinsic surface area for the electrodeposition of CuO, which forms a novel hybrid electrode. The NPG–CuO micro-supercapacitor exhibits maximum areal capacitance 26 mF cm−2, maximum specific energy 3.6 μW h cm−2 and maximum specific power 646 μW cm−2. NPG–CuO micro-supercapacitors show excellent cyclic stability with 98% capacitance retention after 10 000 cycles.

Seminormality and cohomology of projective varieties

Let X be a reduced closed subscheme of p; = Proj R = Y (k a field and R = k[X,,,..., A’,], r 2 2). Let A be the homogeneous co-ordinate ring of X. In [2, 3, 51 the various authors have attempted to determine whether A is CohenMacaulay, or seminormal (or more generally, to study the seminormalization or Cohen-Macaulification of A). The papers mentioned above work primarily with unions of straight lines. In [S, Corollary 5.91 it is proved, for example, that if X is a connected union of lines which have linearly independent directions at each intersection point, then A is seminormal if and only if A is Cohen-Macaulay. The aim of this paper is to give a better understanding of the seminormalization of A and the relationship between seminormality and CohenMacaulayness hinted at by the above mentioned Corollary 5.9. This we do in Section 2 by interpreting the seminormalization of A in terms of sheaf cohomology. We are then able to give a reasonable generalization (our Corollary 3.5) of [S, Corollary 5.91, and also to resolve several computational problems left open in [2, 31, and [4]. The Hilbert function of a graded ring A will be denoted by HA. That is, HA(i) = dim, Ai. If X= Proj A, the Hilbert polynomial of A is denoted HP(X) in some computations of Section 6.

Remarks on maximally differential prime ideals

In this paper we make some remarks on the recent papers of Seibt [4] and Singh [6] on maximally differential prime ideals. Let A be a Noetherian local ring containing a field k of characteristic zero and let P be a prime ideal of A. Our main remarks concern the relationship between the conditions “P is maximally differential” and “P is maximally differential” (cf. [6, Remark (2.1 l)]). Theorem 2.2 provides a partial answer to the question whether these two conditions are equivalent. The contents of this theorem can be described as follows: First, it is shown that if Der(A) = Der(A)a (see Section 1 for notation), then the condition “P is maximally differential” implies the condition “P is maximally differential” (part (2) of the theorem with k = C!!). Note that the condition Der,(A) = Der,(A)A^ is satisfied if fin,,, is finitely generated as an A-module, which is the case, for example, if A is a localization of an afftne algebra over k. Next, assume that k is a coefficient field of A and that A is a G-ring. Suppose P is maximally differential ouer k, i.e., maximally @-differential for a set @ of k-derivations of A. Then it is shown that if Der,(A) = Der,(A)& then the condition “P is maximally differential over k” is equivalent to a condition on P which does not involve the completion of A (parts (2) and (3) of the theorem). This equivalent condition is rank,,, im(pk(P) 2 Der,(A/P)) 2 dim A/P, where ip,(P) is the set of k-derivations of A leaving P stable and I is the natural map. Under more restrictive assumptions on A this condition is shown to be further equivalent to the surjectivity of t (part (4) of the theorem). 387

The structure of generic subintegrality

AbstractIn order to give an elementwise characterization of a subintegral extension of ℚ-algebras, a family of generic ℚ-algebras was introduced in [3]. This family is parametrized by two integral parameters p ⩾ 0,N ⩾ 1, the member corresponding top, N being the subalgebraR = ℚ [{γn¦n ⩾ N}] of the polynomial algebra ℚ[x1,…,xp, z] inp + 1 variables, where