Ziegler spectrum
In mathematics, the Ziegler spectrum of a ring R is a topological space whose points are indecomposable pure-injective right R-modules. Its closed subsets correspond to theories of modules closed under arbitrary products and direct summands. Ziegler spectra are named after Martin Ziegler, who first defined and studied them in 1984.
Definition
Let R be a ring. A pp-n-formula is a formula in the language of R-modules of the formwhere are natural numbers, is an matrix with entries from R, and is an -tuple of variables and is an -tuple of variables.
The Ziegler spectrum,, of R is the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted by, and the topology
has the sets
as subbasis of open sets, where range over
pp-1-formulae and denotes the subgroup of consisting of all elements that satisfy the one-variable formula. One can show that these sets form a basis.
Properties
Ziegler spectra are rarely Hausdorff and often fail to have the -property. However they are always compact and have a basis of compact open sets given by the sets where are pp-1-formulae.When the ring R is countable is sober. It is not currently known if all Ziegler spectra are sober.