This chapter is devoted to reporting on what is known about Ziegler spectra of various types of ring. In some cases results have merely been collected together and stated; in other cases outline or sample proofs are given.
Over some rings, for example, tame hereditary artin algebras and some string algebras, a complete description of the spectrum has been obtained; over others, for example, generalised Weyl algebras, there are not even any isolated points and little is said about the overall structure of the space. In this, and other, “wild” cases, for example, over the Lie algebra sl2(k) and over general pullback rings, it is possible to say something about parts of the space. Over certain rings, the topology on the spectrum is trivial despite there being many points.
Spectra of artin algebras
We begin, in Section 8.1.1, with some observations and open questions. Section 8.1.2 is devoted to describing the Ziegler spectra of tame hereditary finite-dimensional algebras; Section 8.1.3 the spectra of some domestic string algebras. Pure-injectives over the canonical algebras are discussed briefly in Section 8.1.4.
Points of the spectrum
Representations of finite-dimensional algebras have been a rich source of examples, conjectures and results in the application of model theory to modules. After abelian groups, and modules over commutative Dedekind domains, already quite well explored by the early 1970s (for example, [669], [172], also [330], [198]), this was the context where the potential of the interaction was recognised.