On the Chern Number of an Ideal

We settle the negativity conjecture of Vasconcelos for the Chern number of an ideal in certain unmixed quotients of regular local rings by explicit calculation of the Hilbert polynomials of all ideals generated by systems of parameters. Introduction Let I be an m-primary ideal in a local ring (R,m) of dimension d. Let H(I, n) = λ(R/I) denote the Hilbert function of I where λ(M) denotes the length of an R-module M. The Hilbert function H(I, n) is given by a polynomial P (I, n) of degree d for large n. It is written in the form P (I, n) = e0(I) ( n + d− 1 d ) − e1(I) ( n+ d− 2 d− 1 ) + · · ·+ (−1)ed(I). If I is generated by a system of parameters, then R is Cohen-Macaulay if and only if e0(I) = λ(R/I). Recently it has been observed by Vasconcelos [7] that the signature of the coefficient e1(I), called the Chern number of I, can be used to characterize Cohen-Macaulay property of R for large classes of rings. In the Yokohama Conference in 2008 Vasconcelos proposed the following: Negativity Conjecture: Let (R,m) be an unmixed, equidimensional local ring which is a homomorphic image of a Cohen-Macaulay local ring. Then R is not Cohen-Macaulay if and only if for any ideal J generated by a system of parameters, e1(J) < 0. Ghezzi, Hong and Vasconcelos [3] settled the conjecture for (1) Noetherian local domains of dimension d ≥ 2, which are homomorphic images of Cohen-Macaulay local rings and for (2) Universally catenary integral domains containing a field. The Negativity Conjecture has been resolved for all unmixed local rings by S. Goto recently. 2000 Mathematics Subject Classification. 13D40,13D07,13H05.