Let A be a mild k-algebra over an algebraically closed field k, i.e. A is representation-finite or distributive of minimal representation-infinite type. Let S be a simple A-module of finite projective dimension. We establish the strong no loop conjecture for A, which claims that S has only split self-extensions, i.e. the quiver of A has no loop at the vertex corresponding to S. More generally we show that only a small neighborhood of the support of the projective cover of S has to be mild.
Furthermore some reduction techniques are developed for the stronger no loop conjecture and the finitistic dimension conjecture.