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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.

13D05 Homological dimension and commutative rings
13F30 Valuation rings
13G05 Integral domains
Full Text: EuDML
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