We introduce support varieties for rational representations of a linear algebraic group $G$ of exponential type over an algebraically closed field $k$ of characteristic $p>0$. These varieties are closed subspaces of the space $V(G)$ of all 1-parameter subgroups of $G$. The functor $M\mapsto V(G)_{M}$ satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between $V(G)_{M}$ and the family of support varieties $V_{r}(G)_{M}$ obtained by restricting the $G$ action to Frobenius kernels $G_{(r)}\subset G$. These support varieties seem particularly appropriate for the investigation of infinite-dimensional rational $G$-modules.

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.