From associated graded modules to blowing-ups of generalized Cohen-Macaulay modules
A denotes a local ring with maximal ideal \(m\) and M a generalized Cohen- Macaulay module over A, i.e. \(d:=\dim (M)>1\) and \(\ell (H^ i_{{\mathfrak m}}(M))<\infty\) for \(i=0,...,d-1\), where \(H^ i_{{\mathfrak m}}(M)\) denotes the i-th local cohomology module of M with respect to \({\mathfrak m}.\) Let I(M) denote the maximum of the differences I(\({\mathfrak q}:M):=\ell (M/{\mathfrak q}M)-e({\mathfrak q};M)\), where \({\mathfrak q}\) runs through all parameter ideals of M and e(\({\mathfrak q};M)\) is the multiplicity of M with respect to \({\mathfrak q}\). Then M being a generalized Cohen-Macaulay module just means that \(I(M)<\infty.\) If \({\mathfrak q}=(a_ 1,...,a_ d)\) and I(\({\mathfrak q};M)=I(M)\), we call \(a_ 1,...,a_ d\) a standard system of parameters of M. It is known that there exist ideals \({\mathfrak a}\) of A such that every system of parameters of M contained in \({\mathfrak a}\) is standard. Such ideals are called M-standard ideals. In particular, M is a Buchsbaum module if and only if \({\mathfrak m}\) is a M-standard ideal. We compute the local cohomology modules of the associated graded module \(G_{{\mathfrak a}}(M):=\oplus^{\infty}_{n=1}{\mathfrak a}^ nM/{\mathfrak a}^{n+1}M\quad and\) of the Rees module (blowing-up) \(R_{{\mathfrak a}}(M):=\oplus^{\infty}_{n=1}{\mathfrak a}^ nM\quad with\) respect to the maximal graded ideal of \(G_{{\mathfrak a}}(A)\) and \(R_{{\mathfrak a}}(A)\). Sometime, we will imagine of \(R_{{\mathfrak a}}(A)\) as the subring A[\({\mathfrak a}T]\) of A[T], where T is some indeterminate. In particular, since \(G_{{\mathfrak a}}(A)\cong A[{\mathfrak a}T]/{\mathfrak a}A[{\mathfrak a}T]\), we may consider \(G_{{\mathfrak a}}(A)\) and hence \(G_{{\mathfrak a}}(M)\) as a graded \(R_{{\mathfrak a}}(A)\)-module. Let P denote the maximal graded ideal of \(R_{{\mathfrak a}}(M)\) and Q the ideal (\({\mathfrak a},{\mathfrak a}T)\) of \(R_{{\mathfrak a}}(M)\). Main result: Theorem 1.1. Suppose that there exists an integer \(r\leq d-1\) such that \([H^ i_ P(G_{{\mathfrak a}}(M)]_ n=0\) for \(n\neq r-i\) if \(i=0,...,d- 1\), and for \(n>r-d\) if \(i=d\). Then \(R_{{\mathfrak a}}(M)\) is a generalized Cohen-Macaulay module and Q is an \(R_{{\mathfrak a}}(M)\)-standard ideal. Corollary 1.2. Let \(a_ 1,...,a_ d\) be a standard system of parameters of M and \({\mathfrak a}=(a_ 1,...,a_ d)\). Then \(a_ 1,a_ 2-a_ 1T,...,a_ d-a_{d-1}T,a_ dT\), is a standard system of parameters of \((R_{{\mathfrak a}}(M))_ P.\) Corollary 1.3. Let A be a Buchsbaum ring with maximal embedding dimension. Then the Rees algebra \(R_{{\mathfrak m}}(A)\) is also a Buchsbaum ring. Corollary 1.4. Suppose that A is a Buchsbaum ring of the form R/I, where R is a regular local ring with maximal ideal \({\mathfrak n}\) and I is some ideal of R. Then \(R_ m(A)\) is a Buchsbaum ring if there exist some positive integers \(r\leq d-1\) such that \(I\subset {\mathfrak n}^ r\) and \({\mathfrak m}^{r+1}={\mathfrak qm}^ r\) for some parameter ideal \({\mathfrak q}\) of A.
scientific article; zbMATH DE number 4021305