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Functorial finite subcategories over triangular matrix rings


Author: S. O. Smalø
Journal: Proc. Amer. Math. Soc. 111 (1991), 651-656
MSC: Primary 16D90; Secondary 16D20, 16P20, 18A25
DOI: https://doi.org/10.1090/S0002-9939-1991-1028295-9
MathSciNet review: 1028295
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1028295-9
Article copyright: © Copyright 1991 American Mathematical Society

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