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

H. Martin, J. Migliore<P>

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.