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Auslander-Reiten sequences over derivation polynomial rings. (English) Zbl 0742.16008

Let \(F\) be a field with a derivation and \(R\) the corresponding derivation polynomial ring. The article deals with the problem whether the category \({\mathfrak m}_ R\) of all \(R\)-modules of finite length has enough Auslander-Reiten sequences. It was shown by the author [J. Algebra 119, No. 2, 366-392 (1988; Zbl 0661.16024)] that this is true provided \(F\) is the field of formal Laurent series in one variable over a field of characteristic 0. Generalizing the methods developed there the author provides necessary and sufficient conditions for the existence of Auslander-Reiten sequences in \({\mathfrak m}_ R\). Since \(R\)-modules can be treated in the same way like modules over a group algebra, only the Auslander-Reiten sequence ending with the simple \(R\)-module has to be explored. Finally, a class of derivation fields satisfying the conditions is specified. The fields in question are the fields of power series \(k((\Gamma))\), where \(k\) is a field and \(\Gamma\) a linearly ordered group contained in the additive group of \(k\).
Reviewer: H.Meltzer (Berlin)

MSC:

16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16W25 Derivations, actions of Lie algebras
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)

Citations:

Zbl 0661.16024
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References:

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