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Noetherian hereditary abelian categories satisfying Serre duality


Authors: I. Reiten and M. Van den Bergh
Journal: J. Amer. Math. Soc. 15 (2002), 295-366
MSC (2000): Primary 18E10, 18G20, 16G10, 16G20, 16G30, 16G70
DOI: https://doi.org/10.1090/S0894-0347-02-00387-9
Published electronically: January 18, 2002
MathSciNet review: 1887637
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Abstract: In this paper we classify $\operatorname{Ext}$-finite noetherian hereditary abelian categories over an algebraically closed field $k$ satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories.

As a side result we show that when our hereditary abelian categories have no non-zero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.


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Additional Information

I. Reiten
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
Email: idunr@math.ntnu.no

M. Van den Bergh
Affiliation: Department WNI, Limburgs Universitair Centrum, Universitaire Campus, Building D, 3590 Diepenbeek, Belgium
Email: vdbergh@luc.ac.be

DOI: https://doi.org/10.1090/S0894-0347-02-00387-9
Keywords: Noetherian hereditary abelian categories, Serre duality, saturation property
Received by editor(s): December 6, 2000
Published electronically: January 18, 2002
Additional Notes: The second author is a senior researcher at the Fund for Scientific Research. The second author also wishes to thank the Clay Mathematics Institute for material support during the period in which this paper was written.
Article copyright: © Copyright 2002 American Mathematical Society

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