Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 1028295
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [AS1] M. Auslander and S. O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), 61-122. MR 591246 (83a:16039)
  • [AS2] -, Almost split sequences in subcategories, J. Algebra 96 (1981), 426-454. MR 617088 (82j:16048a)
  • [FGR] R. M. Fossum, P. A. Griffith and I. Reiten, Trivial extensions of Abelian categories, Lecture Notes in Math., vol. 456, Springer-Verlag, 1975. MR 0389981 (52:10810)
  • [IST] K. Igusa, S. O. Smalø, and G. Todorov, Finite projectivity and contravariant finiteness, Proc. Amer. Math. Soc 109 (1990), 937-941. MR 1027094 (91b:16010)
  • [PS] J. A. de la Peña and D. Simson, Prinjective modules, reflection functors, quadratic forms and Auslander-Reiten sequences, preprint, 1989.
  • [Gr] R. Grecht, Kategorien von Moduln mit Untermoduln, Diplomarbeit, Zürich, 1986.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16D90, 16D20, 16P20, 18A25

Retrieve articles in all journals with MSC: 16D90, 16D20, 16P20, 18A25

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society