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

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 kx 0 ; : : :; x n ], and the ideals I and J deene 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 deened by I and J. By doing this, we are able to prove several general algebraic results about the deening ideals of certain subschemes of projective space. Our main technique in this paper is a study of the deeciency modules of a subscheme V of P n. These modules are important algebraic invariants of V , and reeect many of the properties of V , both geometric and algebraic. For instance, when V equidimensional and dimV 1, the deeciency modules of V are invariant (up to a shift in grading) along the even liaison class of V ((13], 10], 14], 6]), although they do not in general completely determine the even liaison class, except in the case of curves in P 3 , 13]. On the algebraic side, at least for curves in P 3 , the deeciency modules have been shown to have connections to the number and degrees of generators of the saturated ideal deening V , 11]. 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 rst section, we set up our notation and give the basic deenitions which we will use. Then we prove our main technical result: if I and J deene 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 diierent proof of a general statement due to Serre about when there is an equality of intersection and product. 1 In the second section, we give an extension of Dubreil's Theorem on the number of generators of ideals in a polynomial ring. Speciically, our generalization works for an …