×

On the dimension filtration and Cohen-Macaulay filtered modules. (English) Zbl 0942.13015

Van Oystaeyen, Freddy (ed.), Commutative algebra and algebraic geometry. Proceedings of the Ferrara meeting in honor of Mario Fiorentini on the occasion of his retirement, Ferrara, Italy. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 206, 245-264 (1999).
Let \((A,m)\) be a local (Noetherian) ring and \(M\) a finitely generated \(A\)-module with \(\dim_AM=d\). For each integer \(i\), \(0\leq i\leq d\), let \(M_i\) be the largest submodule of \(M\) with \(\dim_AM_i\leq i\). These submodules form an increasing family of submodules of \(M\) which the author calls the dimension filtration of \(M\). A characterization of the \(M_i\)’s in terms of a normal decomposition of \(O_M\) is given. It is shown that \(M\) has a “distinguished system of parameters” \(x_1,\dots,x_d\) in the sense that \((x_{i+1}, \dots, x_d) M_i=0\) for all \(i=0, \dots, d-1\). If \(A\) possesses a dualizing complex, then the dimension filtration occurs as the filtration of a spectral sequence related to duality. The module \(M\) is defined to be a Cohen-Macaulay filtered module (CMF module) if \(M_i/M_{i-1}\) is either zero or an \(i\)-dimensional Cohen-Macaulay module for each \(i\). A number of basic properties of CMF modules are investigated.
For the entire collection see [Zbl 0913.00044].

MSC:

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13C14 Cohen-Macaulay modules
PDFBibTeX XMLCite