The cohomology ring of truncated quiver algebras

G. AMES, L. CAGLIERO, P. TIRAO

Valladolid, España

Congreso; VASBI, ICM Satellite Conference on K-theory and Noncommutative Geometry.; 2006

In this talk we present a joint work with G. Ames and P. Tirao (see math.KT/0603056) in which we explicitly describe the Yoneda product in the Hochschild cohomology  ring $H^*(A,A)$ of an arbitrary truncated quiver  algebra $A$ in terms of a minimal resolution of $A$.It is known that the Yoneda product has a nice  description on the bar resolution of $A$ as the usual cup product. On the other hand, the cohomology groups  has been described by Locateli in terms of classes of pairs of paths using minimal resolutions. Our first result is the explicit construction of comparison morphisms, in both directions, between the two different resolutions. We are then able to give a clear description of  the Yoneda product on the minimal resolution that, in particular, implies that the product of two cohomology classes of odd degree is equal to zero.  Our main theorem extends, to arbitrary truncated quiver algebras, the results obtained by Bardzell,  Locateli and Marco for cyclic truncated quiver algebras. In addition, we show that the ring structure  (of the augmentation ideal) is trivial if the underlying quiver is not an oriented cycle and has neither sinks nor sources.