Submodules of the deficiency modules and an extension of Dubreil's Theorem

H. Martin, J. Migliore

In its most basic form, Dubreil's Theorem states that for an ideal I defining a codimension 2, arithmetically Cohen-Macaulay subscheme of projective n-space, the number of generators of I is bounded above by the minimal degree of a minimal generator plus 1. By introducing a new ideal J which is the complete intersection of n-1 general linear forms, we are able to extend Dubreil's Theorem to an ideal I defining a locally Cohen-Macaulay subscheme V of any codimension. Our new bound involves the lengths of the Koszul homologies of the cohomology modules of V, with respect to the ideal J, and depends on a careful identification of the module (I \cap J)/IJ in terms of the maps in the free resolution of J. As a corollary to this identification, we also give a new proof of a theorem of Serre which gives a necessary and sufficient condition to have the equality I \cap J = IJ in the case where I and J define disjoint schemes in projective space.