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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let and be Artin algebras, a -bimodule, and the triangular matrix ring of , and ; assume that is also an Artin algebra. The -modules are triples where is a -module, is a -module, and is a -homomorphism from to . For an Artin algebra , let denote the category of finitely generated -modules. For full subcategories of and of , let denote the full subcategory consisting of the modules , where is in and is in . In this paper it is proved that is functorially finite in if and only if is functorially finite in and is functorially finite in . Using this result, we increase the known examples of functorially finite subcategories considerably, hence also the classes of subcategories having relative almost split sequences.

**[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.

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

DOI:
https://doi.org/10.1090/S0002-9939-1991-1028295-9

Article copyright:
© Copyright 1991
American Mathematical Society