Functorial finite subcategories over triangular matrix rings

Abstract:

Let $\Lambda$ and $\Gamma$ be Artin algebras, $M$ a $\Gamma - \Lambda$-bimodule, and $R$ the triangular matrix ring of $\Lambda ,\Gamma$, and $M$; assume that $R$ is also an Artin algebra. The $R$-modules are triples $(U,V,f)$ where $U$ is a $\Lambda$-module, $V$ is a $\Gamma$-module, and $f$ is a $\Gamma$-homomorphism from $M \otimes U$ to $V$. For an Artin algebra $S$, let $\operatorname {mod} S$ denote the category of finitely generated $S$-modules. For full subcategories $S$ of $\operatorname {mod} \Lambda$ and $T$ of $\operatorname {mod} \Gamma$, let $\operatorname {mod} R_T^S$ denote the full subcategory consisting of the modules $(U,V,f)$, where $U$ is in $S$ and $V$ is in $T$. In this paper it is proved that $\operatorname {mod} R_T^S$ is functorially finite in $\operatorname {mod} R$ if and only if $S$ is functorially finite in $\operatorname {mod} \Lambda$ and $T$ is functorially finite in $\operatorname {mod} \Gamma$. Using this result, we increase the known examples of functorially finite subcategories considerably, hence also the classes of subcategories having relative almost split sequences.