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The structure of balanced big Cohen-Macaulay modules over Cohen-Macaulay rings. (English) Zbl 1374.13017

Big Cohen-Macaulay algebras over regular local rings are flat. A theorem of Lazard states that any flat module can be written as a directed limit of free modules. On the other hand small Cohen-Macaulay modules over regular rings are free. In particular, over regular rings, any big Cohen-Macaulay is a direct limit of small Cohen-Macaulay modules. The main aim of this paper is to extend this observation to the class of Cohen-Macaulay local rings. As another result of the paper: “Every finitely generated module has a pre-envelope with respect to the class of finitely generated maximal Cohen-Macaulay modules.”

MSC:

13C14 Cohen-Macaulay modules
13D05 Homological dimension and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
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