Anna Guerrieri

The Dedekind-Mertens formula and determinantal rings

We give a combinatorial proof of the Dedekind–Mertens formula by computing the initial ideal of the content ideal of the product of two generic polynomials. As a side effect we obtain a complete classification of the rank 1 Cohen–Macaulay modules over the determinantal rings K[X]/I2(X). Let f, g be polynomials in one indeterminate over a commutative ring A. The Dedekind-Mertens formula relates the content ideals of f , g, and their product fg: one has c(fg)c(f) = c(g)c(f), d = deg g. It is the best universally valid variant of Gaus’ classical formula c(fg) = c(f)c(g) for polynomials over a principal ideal domain. (The content ideal of f ∈ A[T ] is the ideal generated by the coefficients of f in A.) Content ideals and the Dedekind– Mertens formula have recently received much attention; see Glaz and Vasconcelos [8], Corso, Vasconcelos, and Villarreal [6] and Heinzer and Huneke [9], [10]. For detailed historical information about the Dedekind–Mertens formula, see [9]. The main objective of this paper is a combinatorial proof of the formula based on a Grobner basis approach to the ideal c(fg) for polynomials with indeterminate coefficients; in fact we will determine the initial ideal of c(fg) with respect to a suitable term order. (For information on term orders and Grobner bases we refer the reader to Eisenbud [7].) A side effect of our approach is very precise numerical information about the rank one Cohen–Macaulay modules over the determinantal ring S = K[X ]/I2(X) where X is an m × n matrix of indeterminates and I2(X) the ideal generated by its 2-minors. This connection extends the ideas of [6] and was in fact suggested by them. The actual motive for our work was the need for some explicit computation modulo c(fg) in Boffi, Bruns, and Guerrieri [2], or, more precisely, modulo an ideal generalizing c(fg) slightly. Theorem 1. Let K be a field, R = K[Y1, . . . , Ym, Z1, . . . , Zn] and set dk = ∑ i+j=k uijYiZj, k = 2, . . . , m + n, Received by the editors January 22, 1997 and, in revised form, June 16, 1997. 1991 Mathematics Subject Classification. Primary 13C40, 13C14, 13D40, 13P10.

INVARIANTS OF IDEALS HAVING PRINCIPAL REDUCTIONS

For a regular ideal having a principal reduction in a Noetherian ring we consider the structural numbers that arise from taking the Ratliff–Rush closure of the ideal and its powers. In particular, we analyze the interconnections among these numbers and the relation type and reduction number of the ideal. We prove that certain inequalites hold in general among these invariants, while for ideals contained in the conductor of the integral closure of the ring we obtain sharper results that do not hold in general. We provide applications to the one-dimensional local setting and present a sequence of examples in this context.

Permanental ideals of Hankel matrices

We provide the Gröbner basis and the primary decomposition of the ideals generated by 2 × 2 permanents of Hankel matrices.

Searching for Cutkosky's example

We provide a concrete class of rings in which there exists a primary ideal with respect to the maximal ideal that has only one Rees valuation.