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\(p\)-standard systems of parameters and \(p\)-standard ideals in local rings. (English) Zbl 0987.13014

From the introduction: Let \((A,{\mathfrak m})\) be a commutative Noetherian local ring and \(M\) a finitely generated \(A\)-module with \(\dim M=d\). Let \({\mathfrak q}\) be a parameter ideal of \(M\). It is well-known that the difference between the length and the multiplicity of \({\mathfrak q}\) \[ I({\mathfrak q};M)= \ell(M/{\mathfrak q}M)-e({\mathfrak q};M) \] gives a lot of information on the structure of the module \(M\). We set \(I(M) =\sup_{\mathfrak q}I({\mathfrak q};M)\), where \({\mathfrak q}\) runs through all parameter ideals of \(M\). Then \(I(M)<+\infty\) if and only if \(\ell(H^i_{\mathfrak m}(M))<+ \infty\) for \(i=0,\dots, d-1\), where \(H^i_{\mathfrak m}(M)\) is the \(i\)-th local cohomology module of \(M\) with respect to \({\mathfrak m}\). Modules \(M\) with \(I(M)<+ \infty\) are called generalized Cohen-Macaulay. Recall that a standard system of parameters \(z=\{x_1, \dots,x_d\}\) of \(M\) is characterized by the equality \(I(M)= I({\mathfrak q};M)\), where \({\mathfrak q}=(x_1, \dots,x_d)A\). Then \(M\) is a generalized Cohen-Macaulay module if and only if \(M\) admits a standard system of parameters. Now, we will extend the above idea to the following situation:
Let \(x=\{x_1,\dots, x_d\}\) be a system of parameters of \(M\) and \(n=(n_1, \dots,n_d)\) a \(d\)-tuple of positive integers. We consider the difference \[ I(n; x)=\ell\bigl( M/(x_1^{n_1}, \dots,x_d^{n_d}) \bigr)-n_1\dots n_d e(x;M) \] as a function in \(n\). We proved [Nguyen Tu Cuong, Nagoya Math. J. 125, 105-114 (1992; Zbl 0783.13020)] that the least degree of all polynomials in \(n\) bounding above \(I(n;x)\) is independent of the choice of \(x\). This invariant is called the polynomial type of \(M\) and denoted by \(p(M)\). Therefore \(M\) is not generalized Cohen-Macaulay if and only if \(p(M)>0\). The purpose of this paper is to define a new kind of system of parameters called \(p\)-standard system of parameters, which is closely related to the invariant \(p(M)\) and plays a role in \(M\) with \(p(M)>0\) like that of standard systems of parameters in the theory of generalized Cohen-Macaulay modules.

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C14 Cohen-Macaulay modules

Citations:

Zbl 0783.13020
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