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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Exact categories and vector space categories


Authors: Peter Dräxler, Idun Reiten, Sverre O. Smal\o{}, Øyvind Solberg and with an appendix by B. Keller
Journal: Trans. Amer. Math. Soc. 351 (1999), 647-682
MSC (1991): Primary 16B50; Secondary 16G20, 16G70
MathSciNet review: 1608305
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Abstract: In a series of papers additive subbifunctors $F$ of the bifunctor $\operatorname{Ext}_{\Lambda} (\ ,\ )$ are studied in order to establish a relative homology theory for an artin algebra $\Lambda$. On the other hand, one may consider the elements of $F(X,Y)$ as short exact sequences. We observe that these exact sequences make $\operatorname{mod}\Lambda$ into an exact category if and only if $F$ is closed in the sense of Butler and Horrocks.

Concerning the axioms for an exact category we refer to Gabriel and Roiter's book. In fact, for our general results we work with subbifunctors of the extension functor for arbitrary exact categories.

In order to study projective and injective objects for exact categories it turns out to be convenient to consider categories with almost split exact pairs, because many earlier results can easily be adapted to this situation.

Exact categories arise in representation theory for example if one studies categories of representations of bimodules. Representations of bimodules gained their importance in studying questions about representation types. They appear as domains of certain reduction functors defined on categories of modules. These reduction functors are often closely related to the functor $\operatorname{Ext}_{\Lambda}(\ ,\ )$ and in general do not preserve at all the usual exact structure of $\operatorname{mod}\Lambda$.

By showing the closedness of suitable subbifunctors of $\operatorname{Ext}_{\Lambda}(\ ,\ )$ we can equip $\operatorname{mod}\Lambda$ with an exact structure such that some reduction functors actually become `exact'. This allows us to derive information about the projective and injective objects in the respective categories of representations of bimodules appearing as domains, and even show that almost split sequences for them exist.

Examples of such domains appearing in practice are the subspace categories of a vector space category with bonds. We provide an example showing that existence of almost split sequences for them is not a general fact but may even fail if the vector space category is finite.


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Additional Information

Peter Dräxler
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Email: draexler@mathematik.uni-bielefeld.de

Idun Reiten
Affiliation: Institutt for matematikk og statistikk\ Norges teknisk-naturvitenskapelige universitet\ N-7055 Dragvoll\ Norway
Email: idunr@matsh.ntnu.no

Sverre O. Smal\o{}
Affiliation: Institutt for matematikk og statistikk\ Norges teknisk-naturvitenskapelige universitet\ N-7055 Dragvoll\ Norway
Email: sverresm@math.ntnu.no

Øyvind Solberg
Affiliation: Institutt for matematikk og statistikk\ Norges teknisk-naturvitenskapelige universitet\ N-7055 Dragvoll\ Norway
Email: oyvinso@math.ntnu.no

with an appendix by B. Keller
Affiliation: U.F.R. de Mathématiques, U.R.A. 748 du CNRS, Université Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France
Email: keller@math.jussieu.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02322-3
PII: S 0002-9947(99)02322-3
Received by editor(s): January 27, 1997
Additional Notes: Professors Reiten, Smalø\ and Solberg thank the Norwegian Research Council for partial support during the preparation of this paper.
Article copyright: © Copyright 1999 American Mathematical Society