Distinguishing Leavitt algebras among Leavitt path algebras of finite graphs by the Serre property

Two unanswered questions in the heart of the theory of Leavitt path algebras are whether the Grothendieck group K is a complete invariant for the class of unital purely infinite simple algebras and, a weaker question, whether L2 (the Leavitt path algebra associated to a vertex with two loops) and its Cuntz splice algebra L2 - 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 L2≅ L2 - . Along the way, we give new easy methods to construct algebras having stably free but not free modules.