The usefulness of the Koszul complex in handling in an algebraic setting the two geometric notions of multiplicity and depth first became apparent with the work of Auslander and Buchsbaum [1] following a suggestion of Serre. Regarding the generators a1, …, an of the complex as a 1×n matrix first Eagon and Northcott [4] extended this work to a complex associated with an m×n matrix, then shortly afterwards a different extension was given by Buchsbaum and Rim [2, 3]. These two complexes are two of an infinite family [6] some of which inherit the depth sensitive property of the Koszul complex and all of which under a certain finiteness condition provide the same multiplicity as Euler–Poincaré characteristic [7].

These two properties prove useful in geometric applications, see for example Lago and Rodicio [7] for depth sensitivity and Kirby [8] for the characteristic as multiplicity of intersection. During the course of these developments it became clear that it was most appropriate to regard the complexes as being generated by linear forms. From this point of view it is natural to ask if the linearity of the forms is necessary. In the present note we begin a response to this question by extending the work of [6] to complexes associated with forms of arbitrary positive degree. In a sequel we shall similarly extend the results of multiplicity in [7].