On the Dimension Filtration and Cohen-macaulay Filtered Modules
نویسندگان
چکیده
For a finitely generated A-module M we define the dimension filtration M = {Mi}0≤i≤d, d = dimA M, where Mi denotes the largest submodule of M of dimension ≤ i. Several properties of this filtration are investigated. In particular, in case the local ring (A,m) possesses a dualizing complex, then this filtration occurs as the filtration of a spectral sequence related to duality. Furthermore, we call an A-module M a Cohen-Macaulay filtered module provided all of the quotient modules Mi/Mi−1 are either zero or i-dimensional Cohen-Macaulay modules. We describe a few basic properties of these kind of generalized Cohen-Macaulay modules. In the case A posesses a dualizing complex it turns out – as one of the main results – thatM is a Cohen-Macaulay filtered A-module if and only if for all 0 ≤ i < d the module of deficiency Ki(M) is either zero or an i-dimensional Cohen-Macaulay module. Furthermore basic properties of Cohen-Macaulay filtered modules with respect to localizations, completion, passing to a non-zero divisor, flat extensions are investigated.
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