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Auslander-Reiten sequences over Artinian rings. (English) Zbl 0661.16024

In the middle part of the paper under review the author presents the following example: Let k be some field of characteristic 0, \(F=k((X))\) the field of formal Laurent series in X, \({}_ FN\) be a two-dimensional left vector-space with basis x, y which is made a right vector space by \(x\alpha =\alpha x\), \(y\alpha =\alpha y+\alpha 'x\) (\(\alpha\in F\), ’ the formal derivation) and let \(R=\left( \begin{matrix} F\\ 0\end{matrix} \begin{matrix}_ FN_ F\\ F\end{matrix} \right)\) be the bimodule algebra. By C. M. Ringel [J. Algebra 41, 269-302 (1976; Zbl 0338.16011), Theorem 4 and §§6, 7] the indecomposable finitely presented modules are classified; if \(C\in mod R\) is indecomposable then either \(\partial (C)\neq 0\) (\(\partial\) the defect) which implies dim C\(=(m,n)\) with \(| m-n| =1\) or C is in the abelian category \({\mathfrak m}\times {\mathfrak u}\) of regular modules.
The surprising result (Theorem 13) of section 2 is the following: For each finitely presented, indecomposable, non projective right R-module C there exists an Auslander-Reiten sequence \(0\to A\to B\to C\to 0\) in mod R. This is also an Auslander-Reiten sequence in Mod R if and only if C is not in \({\mathfrak m}.\)
For the proof the author uses his general results on Auslander-Reiten sequences of section 1, f.e. “Theorem 1: Let R be a semiperfect ring and \(C_ R\) a finitely presented, non-projective module with a local endomorphism ring. There exists an Auslander-Reiten sequence \(0\to A\to B\to C\to 0\) in mod R if and only if (Tr C)\({}^ o\) contains a finitely presented, pure submodule with a local endomorphism ring. Up to isomorphism, the latter is uniquely determined”. For the application of these results to the described example the author needs results on systems of differential equations over the field F. These results are proved in section 3 of the paper.
Reviewer: O.Kerner

MSC:

16Gxx Representation theory of associative rings and algebras
16P10 Finite rings and finite-dimensional associative algebras
16P20 Artinian rings and modules (associative rings and algebras)
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)

Citations:

Zbl 0338.16011
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References:

[1] Auslander, M.; Reiten, I., Representation theory of artin algebras. IV. Invariants given by almost split sequences, Comm. Algebra, 5, 443-518 (1977) · Zbl 0396.16007
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