HEIGHTS OF IDEALS OF MINORS By DAVID EISENBUD, CRAIG HUNEKE, and BERND ULRICH

We prove new height inequalities for determinantal ideals in a regular local ring, or more generally in a local ring of given embedding codimension. Our theorems extend and sharpen results of Faltings and Bruns. Introduction. Let φ be a map of vector bundles on a variety X. A wellknown theorem of Eagon and Northcott [EN] gives an upper bound for the codimension of the locus where φ has rank ≤ s for any integer s. Bruns [B] improved this result by taking into account the generic rank r of φ. We shall see below that unlike the Eagon-Northcott estimate, in most cases Bruns’ theorem is sharp only when X is singular. The first goal of this paper is to give stronger results when X is nonsingular, and a little more generally. Strengthening the Eagon-Northcott estimate in a different way from Bruns, Faltings [F] gave an improved bound for the case s = r − 1 under the additional assumption that X is nonsingular and the cokernel of φ is torsion free. We also improve Faltings’ theorem to a result valid for all s. Let R be a ring, and let φ: Rm → Rn be a matrix of rank r. We write Ii = Ii(φ) for the ideal generated by the i × i minors of φ, and we assume that i ≤ r and Ii = R. Bruns’ theorem says that height(Ii) ≤ (r − i + 1)(m + n − r − i + 1). This formula is sharp for every m, n, r, i: take φ to be the image of the generic n × m matrix Φ = (xij) 1 ≤ i ≤ n, 1 ≤ j ≤ m over the ring R = k[{xij}]/Ir+1(Φ). Note that this ring is singular for 0 < T < min{m, n}. For simplicity, for the remainder of this introduction we consider the case in which the ring is regular. Under this hypothesis we give a bound which in general improves Bruns’ bound as follows: Manuscript received May 13, 2002. Research supported in part by the NSF. American Journal of Mathematics 126 (2004), 417–438.