A note on the coefficients of Hilbert characteristic functions in semi-regular local rings

Abstract

Let Q be a semi-regular local ring of dimension d , m be its maximal ideal, and q be an m -primary ideal. Then L Q ( Q / q n +1 ), the length of Q -module Q / q n +1 , is equal to the characteristic polynomial P Q ( q , n ) in n for a sufficiently large value of n : where e i = e i ( q ), i = 0,1,2,…, d are integers uniquely determined by q , called normalized Hilbert coefficients of q according to (1). It was shown in (1) that e 1 ( q ) is a non-negative integer, and is equal to zero if and only if q is generated by a system of parameters. We shall prove, in this paper, that e 2 ( q ) is also a non-negative integer, and that this non-negativity is not necessarily true for other coefficients. We shall give an example with negative e 3 ( q ).