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

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.