Submodules of the Deficiency Modules and an Extension of Dubreil's Theorem

Abstract

In this paper, we consider a basic question in commutative algebra: if I and J are ideals of a commutative ring S , when does IJ = I ∩ J ? More precisely, our setting will be in a polynomial ring k [ x 0 , …, x n ], and the ideals I and J will define subschemes of the projective space ℙ n k over k . In this situation, we are able to relate the equality of product and intersection to the behavior of the cohomology modules of the subschemes defined by I and J . By doing this, we are able to prove several general algebraic results about the defining ideals of certain subschemes of projective space. Our main technique in this paper is a study of the deficiency modules of a subscheme V of ℙ n . These modules are important algebraic invariants of V , and reflect many of the properties of V , both geometric and algebraic. For instance, when V is equidimensional and dim V [ges ]1, the deficiency modules of V are invariant (up to a shift in grading) along the even liaison class of V [ 14, 11, 15, 7 ], although they do not in general completely determine the even liaison class, except in the case of curves in ℙ 3 [ 14 ]. On the algebraic side, at least for curves in ℙ 3 , the deficiency modules have been shown to have connections to the number and degrees of generators of the saturated ideal defining V [ 12 ]. One of our main goals in this paper is to extend these results to subschemes of arbitrary codimension in any projective space ℙ n . We now describe the contents of this paper more precisely. In the first section, we set up our notation and give the basic definitions which we will use. Then we prove our main technical result: if I and J define subschemes V and Y , respectively, of ℙ n , we relate the quotient module ( I ∩ J )/ IJ to the cohomology of V , at least when V and Y meet properly. We are then able to give a different proof of a general statement due to Serre about when there is an equality of intersection and product. In the second section, we give an extension of Dubreil's Theorem on the number of generators of ideals in a polynomial ring. Specifically, our generalization works for an ideal I defining a locally Cohen–Macaulay, equidimensional subscheme V of any codimension in ℙ n , and relates the number of generators of the defining ideal to the length of certain Koszul homologies of the cohomology of V . The results in this section depend crucially on the identification done in Section 1 of the intersection modulo the product. Finally, in Section 3, we give an extension of a surprising result of Amasaki [ 1 ] showing a lower bound for the least degree of a minimal generator of the ideal of a Buchsbaum subscheme. Originally, Amasaki gave a bound in the case of Buchsbaum curves in ℙ 3 (and later gave a natural extension to Buchsbaum codimension 2 subschemes of ℙ n [ 2 ]). Easier proofs were subsequently given by Geramita and Migliore in [ 6 ], based on a determination of the free resolution of the ideal from a resolution of its deficiency module. For Buchsbaum codimension 2 subschemes of ℙ n whose intermediate cohomology vanishes, we are able to extend these considerations.