## 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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