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)


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)


Zbl 0661.16024
Full Text: DOI


[1] Auslander, M., Functors and morphisms determined by objects, Proceedings of the Philadelphia Conference, New York. Proceedings of the Philadelphia Conference, New York, Representation Theory of Algebras, 1-244 (1978) · Zbl 0383.16015
[2] Auslander, M.; Carlson, J. F., Almost split sequences and group rings, J. Algebra, 103, 122-140 (1986) · Zbl 0594.20005
[3] Hausdorff, H., Grundzüge der Mengenlehre (1914), Leipzig
[4] Johnson, J. L., Extensions of differential modules over formal power series rings, Amer. J. Math., 93, 731-741 (1971) · Zbl 0235.12103
[5] Kaplansky, I., An Introduction to Differential Algebra (1957), Hermann: Hermann Paris · Zbl 0083.03301
[6] McConnell, J. C.; Robson, J. C., Homomorphisms and extensions of modules over certain differential polynomial rings, J. Algebra, 26, 319-342 (1973) · Zbl 0266.16031
[7] Ringel, C. M., Representations of \(K\)-species and bimodules, J. Algebra, 41, 269-302 (1976) · Zbl 0338.16011
[8] Zimmermann, W., Existenz von Auslander-Reiten Folgen, Arch. Math., 40, 40-49 (1983) · Zbl 0513.16019
[9] Zimmermann, W., Auslander-Reiten sequences over artinian rings, J. Algebra, 119, 366-392 (1988) · Zbl 0661.16024
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