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Prinjective modules, reflection functors, quadratic forms, and Auslander- Reiten sequences. (English) Zbl 0789.16010

Summary: Let \(A\), \(B\) be artinian rings and let \(_ AM_ B\) be an (\(A\)-\(B\))- bimodule which is a finitely generated left \(A\)-module and a finitely generated right \(B\)-module. A right \(_ AM_ B\)-prinjective module is a finitely generated module \(X_ R = (X_ A',X_ B^{\prime\prime}\), \(\varphi: X_ A' \otimes_ A M_ B \to X_ B^{\prime\prime})\) over the triangular matrix ring \(R = \left({A\atop 0}{_ AM_ B\atop B}\right)\) such that \(X_ A'\) is a projective \(A\)-module, \(X_ B^{\prime\prime}\) is an injective \(B\)-module, and \(\varphi\) is a \(B\)- homomorphism. We study the category \(\text{prin}(R)^ A_ B\) of right \(_ AM_ B\)-prinjective modules. It is an additive Krull-Schmidt subcategory of \(\text{mod}(R)\) closed under extensions. For every \(X\), \(Y\) in \(\text{prin}(R)^ A_ B\), \(\text{Ext}^ 2_ R(X,Y)=0\). When \(R\) is an Artin algebra, the category \(\text{prin}(R)^ A_ B\) has Auslander-Reiten sequences and they can be computed in terms of reflection functors. In the case that \(R\) is an algebra over an algebraically closed field we give conditions for \(\text{prin}(R)^ A_ B\) to be representation-finite or representation-tame in terms of a Tits form. In some cases we calculate the coordinates of the Auslander-Reiten translation of a module using a Coxeter linear transformation.

MSC:

16G10 Representations of associative Artinian rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
18E10 Abelian categories, Grothendieck categories
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