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
k[x0, …, xn],
and the ideals I and J will define subschemes of
the projective space ℙnk
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 defined by I and J. By doing this,
we are able to prove several general
algebraic results about the defining ideals of certain subschemes of projective
space.
Our main technique in this paper is a study of the deficiency modules
of a
subscheme V of ℙn. These modules are
important algebraic invariants of V, and
reflect many of the properties of V, both geometric and algebraic.
For instance, when
V is equidimensional and dim V[ges ]1, the deficiency
modules of V are invariant (up to
a shift in grading) along the even liaison class of V
[14, 11, 15, 7],
although they do not in general completely determine the even liaison class,
except in the case of curves
in ℙ3 [14]. On the algebraic side, at
least for curves in
ℙ3, the deficiency modules have
been shown to have connections to the number and degrees of generators
of the
saturated ideal defining V [12]. One of our
main
goals in this paper is to extend these
results to subschemes of arbitrary codimension in any projective space
ℙn.
We now describe the contents of this paper more precisely. In the first
section, we
set up our notation and give the basic definitions which we will use. Then
we prove
our main technical result: if I and J define
subschemes V and Y, respectively, of ℙn,
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 different proof of
a general statement due
to Serre about when there is an equality of intersection and product.
In the second section, we give an extension of Dubreil's Theorem
on the number
of generators of ideals in a polynomial ring. Specifically, our generalization
works for
an ideal I defining a locally Cohen–Macaulay,
equidimensional subscheme V of any
codimension in ℙn, and relates the number of
generators of the defining ideal to the
length of certain Koszul homologies of the cohomology of V. The
results in this
section depend crucially on the identification done in Section 1 of the
intersection
modulo the product.
Finally, in Section 3, we give an extension of a surprising result of
Amasaki
[1] showing a lower bound for the least degree of
a minimal generator of the ideal of a
Buchsbaum subscheme. Originally, Amasaki gave a bound in the case of Buchsbaum
curves in ℙ3 (and later gave a natural extension to Buchsbaum
codimension 2
subschemes of ℙn [2]). Easier
proofs were subsequently given by Geramita and
Migliore in [6], based on a determination of the
free resolution of the ideal from a
resolution of its deficiency module. For Buchsbaum codimension 2
subschemes of ℙn
whose intermediate cohomology vanishes, we are able to extend these considerations.