In mathematics, the (right) Ziegler spectrum of a ring R is a topological space whose points are (isomorphism classes of) 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.
[1] Let R be a ring (associative, with 1, not necessarily commutative).
A (right) pp-n-formula is a formula in the language of (right) R-modules of the form where
, 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
consisting of all elements that satisfy the one-variable formula
One can show that these sets form a basis.
Ziegler spectra are rarely Hausdorff and often fail to have the
However they are always compact and have a basis of compact open sets given by the sets
[2] It is not currently known if all Ziegler spectra are sober.
Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherent Grothendieck category, which generalizes the construction above.