Abstract
Two unanswered questions in the heart of the theory of Leavitt path algebras are whether the Grothendieck group \(K_0\) is a complete invariant for the class of unital purely infinite simple algebras and, a weaker question, whether \(L_2\) (the Leavitt path algebra associated to a vertex with two loops) and its Cuntz splice algebra \(L_{2-}\) are isomorphic. A positive answer to the first question implies the latter. In this short paper, we raise and investigate another question, the so-called Serre conjecture, which sits in between of the above two questions: A positive answer to the classification question implies Serre’s conjecture which in turn implies \(L_2 \cong L_{2-}\). Along the way, we give new easy methods to construct algebras having stably free but not free modules.
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Acknowledgements
A part of this work was done when the first author was an Alexander von Humboldt Fellow at the University of Münster in the winter of 2021. He would like to thank both institutions for the excellent hospitality. The authors would like to express their appreciations to the referees for their valuable comments. We would also like to thank Mark Tomforde for his explanations about the Kirchberg-Phillips theorem.
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Hazrat, R., Rangaswamy, K.M. Distinguishing Leavitt algebras among Leavitt path algebras of finite graphs by the Serre property. Arch. Math. 121, 133–143 (2023). https://doi.org/10.1007/s00013-023-01880-z
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DOI: https://doi.org/10.1007/s00013-023-01880-z