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Balanced projective dimension of modules. (English) Zbl 0929.13006
Summary: This semi-expository note fulfills the promise we gave some time ago of building a theory of balanced projective dimension of torsion-free modules over valuation domains, which will be used in further studies of balanced projective dimension. A complementary account of balanced projective dimension for the torsion case (for abelian \(p\)-groups) has been given by Fuchs and Hill. In addition to essential properties of balanced exact sequences, we define balanced \(n\)-fold extensions and define a module \(M\) to have the balanced projective dimension \(n\), if there is a module \(C\) with \(\text{Ext}^n_{\mathcal B}(M,C)\not=0\) and, for every \(k\geq 1\) and every module \(B\), \(\text{Ext}^{n+k}_B(M,B)=0\). Principal results are variations on the theme of the well-known Auslander lemma about the dimension of the union of an ascending chain of submodules depending on the dimension of the subsequent links.

MSC:
13D05 Homological dimension and commutative rings
13F30 Valuation rings
13G05 Integral domains
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References:
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