The strong no loop conjecture is true for mild algebras

Abstract

Let A be a finite dimensional associative algebra over an algebraically closed field with a simple module S of finite projective dimension. The strong no loop conjecture says that this implies Ext(S,S)=0, i.e. that the quiver of A has no loops in the point corresponding to S. In this paper we prove the conjecture in case A is mild, which means that A has only finitely many two-sided ideals and each proper factor algebra A/J is representation finite. In fact, it is sufficient that a "small neighborhood" of the support of the projective cover of S is mild.