Throughout this paper, let M be a finitely generated module over a Noether local
ring (A, [mfr ]) with dim M = d. Let
x = (x1, …, xd)
be a system of parameters of M and n =
(n1, …, nd) ∈ ℕd
a d-tuple of positive integers. This paper is concerned with the
following two points of view.
First, it is well-known that, the difference between lengths and multiplicities
considered as a function in n, gives a lot of information on the structure of M. This
function in general is not a polynomial in n for all
n1, …, nd
large enough (n [Gt ] 0 for short). But, it was shown in [C3] that
the least degree of all polynomials in n
bounding above IM (n, x) is
independent of the choice of x. This numerical invariant
is denoted by p(M) and called the polynomial type of the module M.
By [C2] and [C3]
this polynomial type does not change by the [mfr ]-adic completion Mˆ of M and p(M) is
just equal to the dimension of the non-Cohen–Macaulay locus of Mˆ (see
[C2, C3, C4, CM] for
more details). Therefore a module M is Cohen–Macaulay or
generalized Cohen–Macaulay if and only if
p(M) = − ∞ or p(M) [les ] 0, respectively,
where we set by − ∞ the degree of the zero-polynomial. However, one knows little
about the structure of M when p(M) > 0.
Second, following Sharp and Hamieh ([SH]), we consider the difference
as a function in n, where
is the cyclic submodule of the module of generalized fractions
defined in [SZ1].