The strong no loop conjecture is true for mild algebras

TL;DR

This paper proves the strong no loop conjecture for mild algebras, showing that simple modules with finite projective dimension imply the absence of loops in the algebra's quiver.

Contribution

It establishes the conjecture for mild algebras, expanding understanding of the relationship between projective dimension and quiver structure.

Findings

Proves the strong no loop conjecture for mild algebras.

Shows that finite projective dimension implies no loops in the quiver.

Ext(S,S)=0 for simple modules with finite projective dimension in 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.