Special biserial algebras with no outer derivations
Let A be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of A with coefficients in the bimodule A vanishes if and only if A is representation-finite and simply connected (in the sense of Bongartz and Gabriel), if and only if the Euler characteristic of Q equals the number of indecomposable non-uniserial projective-injective A-modules (up to isomorphism). Moreover, if this is the case, then all the higher Hochschild cohomology groups...