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

Abstract: Two unanswered questions in the heart of the theory of Leavitt path algebras are whether 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. The positive answer to the first question implies the latter. In this short paper, we raise and investigate another question, the so-called Serre's conjecture, which sits in between of the above two questions: The 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 to construct algebras having stable free but not free modules.