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Relations for Grothendieck groups of Gorenstein rings


Author: Naoya Hiramatsu
Journal: Proc. Amer. Math. Soc. 145 (2017), 559-562
MSC (2010): Primary 13D15; Secondary 16G50, 16G60
DOI: https://doi.org/10.1090/proc/13255
Published electronically: August 5, 2016
MathSciNet review: 3577860
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Abstract: We consider the converse of the Butler, Auslander-Reiten Theorem, which concerns the relations for Grothendieck groups. We show that a Gorenstein ring is of finite representation type if the Auslander-Reiten sequences generate the relations for Grothendieck groups. This gives an affirmative answer of the conjecture due to Auslander.


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Additional Information

Naoya Hiramatsu
Affiliation: Department of General Education, Kure National College of Technology, 2-2-11, Agaminami, Kure Hiroshima, 737-8506 Japan
Email: hiramatsu@kure-nct.ac.jp

DOI: https://doi.org/10.1090/proc/13255
Keywords: Grothendieck group, finite representation type, AR sequence
Received by editor(s): January 28, 2016
Received by editor(s) in revised form: April 21, 2016
Published electronically: August 5, 2016
Additional Notes: This work was partly supported by JSPS Grant-in-Aid for Young Scientists (B) 15K17527.
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society