Generalized Hilbert Functions

Let $M$ be a finite module and let $I$ be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of $I$ on $M$ using the 0th local cohomology functor. We show that our definition re-conciliates with that of Ciuperc$\breve{{\rm a}}$. By generalizing Singh's formula (which holds in the case of $\lambda(M/IM)<\infty$), we prove that the generalized Hilbert coefficients $j_0,..., j_{d-2}$ are preserved under a general hyperplane section, where $d={\rm dim}\,M$. We also keep track of the behavior of $j_{d-1}$. Then we apply these results to study the generalized Hilbert function for ideals that have minimal $j$-multiplicity or almost minimal $j$-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal $j$-multiplicity does not have the `expected' shape described in the case where $\lambda(M/IM)<\infty$. Finally we give a sufficient condition such that the generalized Hilbert series has the desired shape.Comment: arXiv admin note: text overlap with arXiv:1101.228

Generalized stretched ideals and Sally Conjecture

We introduce the concept of $j$-stretched ideals in a Noetherian local ring. This notion generalizes to arbitrary ideals the classical notion of stretched $\mathfrak{m}$-primary ideals of Sally and Rossi-Valla, as well as the concept of ideals of minimal and almost minimal $j$-multiplicity introduced by Polini-Xie. One of our main theorems states that, for a $j$-stretched ideal, the associated graded ring is Cohen-Macaulay if and only if two classical invariants of the ideal, the reduction number and the index of nilpotency, are equal. Our second main theorem, presenting numerical conditions which ensure the almost Cohen-Macaulayness of the associated graded ring of a $j$-stretched ideal, provides a generalized version of Sally's conjecture. This work, which also holds for modules, unifies the approaches of Rossi-Valla and Polini-Xie and generalizes simultaneously results on the Cohen-Macaulayness or almost Cohen-Macaulayness of the associated graded module by several authors, including Sally, Rossi-Valla, Wang, Elias, Corso-Polini-Vaz Pinto, Huckaba, Marley and Polini-Xie.Comment: 25 pages (modified the presentation of the material and added examples). Comments are welcom

On the Cohen-Macaulayness of the conormal module of an ideal

In the present paper we investigate a question stemming from a long-standing conjecture of Vasconcelos: given a generically a complete intersection perfect ideal I in a regular local ring R, is it true that if I/I^2 (or R/I^2) is Cohen-Macaulay then R/I is Gorenstein? Huneke and Ulrich, Minh and Trung, Trung and Tuan and - very recently - Rinaldo Terai and Yoshida, already considered this question and gave a positive answer for special classes of ideals. We give a positive answer for some classes of ideals, however, we also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings showing that in general the answer is negative. The homogeneous examples have been found thanks to the help of J. C. Migliore. Furthermore, the counterexamples show the sharpness of our main result. As a by-product, we exhibit several classes of Cohen-Macaulay ideals whose square is not Cohen-Macaulay. Our methods work both in the homogeneous and in the local settings.Comment: 24 pages. Added a few reference

Does the Market Pay Off? Earnings Inequality and Returns to Education in Urban China

The paper examines earnings inequality and earnings returns to education in China among four types of workers characterized by their labor market history. Compared to workers staying in the state sector, early market entrants no longer enjoyed advantages. The commonly observed higher earnings returns to education in the market sector are only limited to recent market entrants. This results from the aggregation of two very different types of workers: those who were "pushed" and those who "jumped" into the market in later stage of the reform. The findings challenge the prevailing wisdom that education is necessarily more highly rewarded by the market sector.http://deepblue.lib.umich.edu/bitstream/2027.42/39838/3/wp454.pd