Nicole Snashall and Øyvind Solberg

We define a support variety for a finitely generated module over an artin algebra $\Lambda$ over a commutative artinian ring $k$, with $\Lambda$ flat as a module over $k$, in terms of the maximal ideal spectrum of the algebra ${\rm HH}^*(\Lambda)$ of $\Lambda$. This is modelled on what is done in modular representation theory, and the varieties defined in this way are shown to have many of the same properties as those for group rings. In fact the notions of a variety in our sense and those for principal and non-principal blocks are related by a finite surjective map of varieties. For a finite-dimensional self-injective algebra over a field, the variety is shown to be an invariant of the stable component of the Auslander--Reiten quiver. Moreover, we give information on nilpotent elements in ${\rm HH}^*(\Lambda)$, give a thorough discussion of the ring ${\rm HH}^*(\Lambda)$ on a class of Nakayama algebras, give a brief discussion of a possible notion of complexity, and make a comparison with support varieties for complete intersections.

2000 Mathematical Subject Classification: 16E40, 16G10, 16P10, 16P20 (primary), 13D03, 16G70, 20J06 (secondary).

Keywords: support variety, Hochschild cohomology, artin algebra, Ext algebra.

E-mail:
njs5@mcs.le.ac.uk
oyvinso@math.ntnu.no