Hochschild Cohomology and Representation-finite Algebras

Abstract

Using Grothendieck's semicontinuity theorem for half-exact functors, we derive two semicontinuity results on Hochschild cohomology. We apply these to show that the first Hochschild cohomogy group of the mesh algebra of a translation quiver over a domain vanishes if and only if the translation quiver is simply connected. We then establish an exact sequence relating the first Hochschild cohomology group of an algebra to that of the endomorphism algebra of a module and apply it to study the first Hochschild cohomology group of an Auslander algebra. Our main result shows that for a finite-dimensional and representation-finite algebra algebra $A$ over an algebraically closed field with Auslander algebra $\Lambda$ the following conditions are equivalent:

(1) $A$ admits no outer derivation;

(2) $\Lambda$ admits no outer derivations;

(3) $A$ is simply connected;

(4) $\Lambda$ is strongly simply connected.