Abstract
We show that every finitely generated conical refinement monoid can be represented as the monoid \(\mathcal V (R)\) of isomorphism classes of finitely generated projective modules over a von Neumann regular ring R. To this end, we use the representation of these monoids provided by adaptable separated graphs. Given an adaptable separated graph (E, C) and a field K, we build a von Neumann regular K-algebra \(Q_K(E,C)\) and show that there is a natural isomorphism between the separated graph monoid M(E, C) and the monoid \(\mathcal V (Q_K(E,C))\).
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Acknowledgements
This research project was initiated when the authors were at the Centre de Recerca Matemàtica as part of the Intensive Research Program Operator algebras: dynamics and interactions in 2017, and continued at the Universidad de Cádiz due to an invitation of the third author to the first two authors in 2018. The work developed was significantly supported by the research environment and facilities provided in both centers. We would also like to thank the anonymous referee for very useful comments and suggestions.
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Dedicated to the memory of Antonio Rosado Pérez
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The three authors were partially supported by the DGI-MINECO and European Regional Development Fund, jointly, through the Grant MTM2017-83487-P. The first and second authors were partially supported by the Generalitat de Catalunya through the Grant 2017-SGR-1725. The second author was partially supported by the Beatriu de Pinós postdoctoral programme of the Government of Catalonia’s Secretariat for Universities and Research of the Ministry of Economy and Knowledge (BP2017-0079). The third author was partially supported by PAI III Grant FQM-298 of the Junta de Andalucía.
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Ara, P., Bosa, J. & Pardo, E. The realization problem for finitely generated refinement monoids. Sel. Math. New Ser. 26, 33 (2020). https://doi.org/10.1007/s00029-020-00559-5
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DOI: https://doi.org/10.1007/s00029-020-00559-5