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

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

1994-10-26

arxiv.org/abs/alg-geom/9410027v1

Mathematics

Algebraic Geometry

18 pages, LaTeX, version 2.09

Scientific paper

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.

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