Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

Abstract

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]/I 2 (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 f g: one has c(f g)c(f) d = c(g)c(f) d+1 , d= deg g. It is the best universally valid variant of Gauß' classical formula c(f g) = 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 Gröbner basis approach to the ideal c(f g) for polynomials with indeterminate coefficients; in fact we will determine the initial ideal of c(f g) with respect to a suitable term order. (For information on term orders and Gröbner 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]/I 2 (X) where X is an m × n matrix of indeterminates and I 2 (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(f g) in Boffi, Bruns, and Guerrieri [2], or, more precisely, modulo an ideal generalizing c(f g) slightly. Theorem 1. Let K be a field, R = K[

Koszul blowup algebras associated to three-dimensional Ferrers diagrams

“…We prove by the induction on a D . When a D = 1, this can be reduced to the twodimensional case in [10,Theorem 3.3]. Now, we assume that a D ≥ 2.…”

“…Since both papers [9] and [10] involve monomial ideals generated in degree two, it is natural to inquire about the degree three case. As a result, we consider the three-dimensional Ferrers diagram and the monomial ideal associated to it.…”

Generalized Newton complementary duals of monomial ideals

“…The work of Corso, Nagel, Petrovic, and Yuen [5] considered the special fiber ring FpI λ´µ q of the specialization of the generalized Ferrers ideal I λ´µ with λ " pλ 1 , . .…”

“…Let D " D λ´µ be a connected shifted quasi-Ferrers diagram with all horizontal moves point westward. Then, D λ´µ is compatible if and only if for every i 1 ă i ă j in rνs, (5) pi 1 , iq, pi, jq P D ùñ pi, j´1q P D.…”

“…The ideals that we consider are duals of specializations of generalized Ferrers ideals. As a corollary of Theorem 3.1 and the work of Corso, Nagel, Petrović, and Yuen in [5], we describe the special fiber rings of the generalized duals of specializations of generalized Ferrers ideals. In other words, we describe the toric rings associated to the generalized duals of such ideals (Corollary 3.3).…”