Lengths of generalized fractions of modules having small polynomial type
Abstract
Throughout this paper, let M be a finitely generated module over a Noether local ring (A, [fraktur m]) with dim M = d. Let x = (x1, …, xd) be a system of parameters of M and <U>n</U> = (n1, …, nd) [set membership] [open face N]d 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 multiplicitiesformula hereconsidered as a function in <U>n</U>, gives a lot of information on the structure of M. This function in general is not a polynomial in <U>n</U> for all n1, …, nd large enough (n [dbl greater-than sign] 0 for short). But, it was shown in [C3] that the least degree of all polynomials in <U>n</U> bounding above IM (<U>n</U>, <U>x</U>) is independent of the choice of <U>x</U>. 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 [fraktur m]-adic completion M^ of M and p(M) is just equal to the dimension of the non-Cohen-Macaulay locus of M^ (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) = [minus sign] [infty infinity] or p(M) [less-than-or-eq, slant] 0, respectively, where we set by [minus sign] [infty infinity] 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 differenceformula hereas a function in <U>n</U>, whereformula hereis the cyclic submodule of the module of generalized fractions U(M)[minus sign]d[minus sign]1d+1M defined in [SZ1].
- Publication:
-
Mathematical Proceedings of the Cambridge Philosophical Society
- Pub Date:
- March 2000
- DOI:
- 10.1017/S0305004199003941
- Bibcode:
- 2000MPCPS.128..269C