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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Prinjective modules, reflection functors, quadratic forms, and Auslander-Reiten sequences

Authors: J. A. de la Peña and D. Simson
Journal: Trans. Amer. Math. Soc. 329 (1992), 733-753
MSC: Primary 16D90; Secondary 16D20, 16G70, 16P20
MathSciNet review: 1025753
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Abstract: Let $ A,\;B$ be artinian rings and let $ _A{M_B}$ be an $ (A - B)$-bimodule which is a finitely generated left $ A$-module and a finitely generated right $ B$-module. A right $ _A{M_B}$-prinjective module is a finitely generated module $ {X_R} = (X_A', X_B'', \varphi :X_A' \otimes_A M_B \to X''_B)$ over the triangular matrix ring

$\displaystyle R = \left( {\begin{array}{*{20}{c}} A & {_A{M_B}} \\ 0 & B \\ \end{array} } \right)$

such that $ X_A'$ is a projective $ A$-module, $ X''_B$ is an injective $ B$-module, and $ \varphi $ is a $ B$-homomorphism.

We study the category $ \operatorname{prin} (R)_B^A$ of right $ _A{M_B}$-prinjective modules. It is an additive Krull-Schmidt subcategory of $ \bmod (R)$ closed under extensions. For every $ X,\;Y$ in $ \operatorname{prin} (R)_B^A,\;\operatorname{Ext} _R^2(X,\,Y) = 0$. When $ R$ is an Artin algebra, the category $ \operatorname{prin} (R)_B^A$ 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 $ \operatorname{prin} (R)_B^A$ 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.

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Article copyright: © Copyright 1992 American Mathematical Society