Abelian categories, almost split sequences, and comodules

Authors:
Mark Kleiner and Idun Reiten

Journal:
Trans. Amer. Math. Soc. **357** (2005), 3201-3214

MSC (2000):
Primary 18E10, 18E20, 16G10, 16G20, 16G30, 16G70, 16W30

DOI:
https://doi.org/10.1090/S0002-9947-04-03571-8

Published electronically:
September 23, 2004

MathSciNet review:
2135742

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The following are equivalent for a skeletally small abelian Hom-finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length.

(a) Each indecomposable injective has a simple subobject.

(b) The category is equivalent to the category of socle-finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle-finitely copresented.

(c) The category has left almost split sequences.

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Additional Information

**Mark Kleiner**

Affiliation:
Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150

Email:
mkleiner@syr.edu

**Idun Reiten**

Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Email:
idunr@math.ntnu.no

DOI:
https://doi.org/10.1090/S0002-9947-04-03571-8

Keywords:
Abelian category,
almost split sequence,
semiperfect cocoherent coalgebra,
comodule

Received by editor(s):
May 2, 2003

Received by editor(s) in revised form:
November 17, 2003

Published electronically:
September 23, 2004

Additional Notes:
The main results were obtained when the first-named author visited Norwegian University of Science and Technology in November–December of 2001. He expresses his sincere gratitude to the members of the Department of Mathematical Sciences for their warm hospitality. The work was finished in February 2003, when the authors participated in the program in Commutative Algebra at the Mathematical Sciences Research Institute, Berkeley. The authors thank the members of the institute for their hospitality.

Dedicated:
Dedicated to the memory of Sheila Brenner

Article copyright:
© Copyright 2004
American Mathematical Society