Special biserial algebras with no outer derivations
Tom 125 / 2011
Colloquium Mathematicum 125 (2011), 83-98 MSC: Primary 16E40, 16G60. DOI: 10.4064/cm125-1-6
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 of $A$ vanish.