Some thirty years ago Buchsbaum and Rim [1]
extended the notion of multiplicity
e(a1, …, an;
E)
for elements a1, …, an
of a commutative ring R with identity and a
Noetherian R-module E (≠0) with
lengthR
(E/[sum ]nt=1aiE) finite to give a multiplicity
e((aij); E)
associated with E and an m×n matrix
(aij) over R satisfying a certain
extended finiteness condition. One of their results states that for each
of a set of m
complexes depending on E, (aij) the
Euler–Poincaré characteristic is a certain integer
multiple of e((aij); E),
at least
when R is a local ring.
Some of these ideas were taken up in [4] where it
is
shown that when n[ges ]m−1,
each of the complexes K((aij); E;
t)
with t∈ℤ introduced in [3] also have
e((aij); E) as
their Euler–Poincaré characteristic. With a slight change
in
viewpoint (aij) can be replaced by linear
forms
aj=[sum ]mi=1aijxi
(j=1, …, n) of the graded polynomial ring
R[x1, …, xm];
the complex K((aij); E;
t)
then becomes the component of degree t in a certain graded double
complex
formula here
where K(a1, …, an;
F) is the standard Koszul complex (see [4; section
2]). From this
point of view the construction can be extended to allow the homogeneous
polynomials
a1, …, an
to have any (possibly unequal) positive degrees [6].
The main aim of the
present note is to extend similarly the results of [4]
and
to strengthen those results
to give information on the vanishing of the multiplicity.