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

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