1 J ul 1 99 9 Artin Algebras with Loops but no Outer

the k-vectorspace of outer k-derivations , that is, the quotient of all k-derivations of Λ modulo the inner ones. It has been suspected for some time, and [4] seems to be the earliest reference, that vanishing of the first Hochschild cohomology precludes the existence of oriented cycles in the ordinary quiver. There are various supporting partial results, among them [8, (2.3), (3.2)], [2, (2.2)], [7, (1.3)]. An algebra Λ without oriented cycles in its ordinary quiver is of finite global dimension. In turn, finite global dimension implies that there are no loops in the ordinary quiver, see [10] or [9]. The following result shows that even the existence of loops in the ordinary quiver is no guarantee for the existence of non trivial outer derivations, thus refuting the above suspicion.