Ziegler spectrum
# Ziegler spectrum

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.

## Definition

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

    $\exists {\overline {y}}\ ({\overline {y}},{\overline {x}})A=0$

where $\ell ,n,m$ are natural numbers, $A$ is an $(\ell +n)\times m$ matrix with entries from R, and ${\overline {y}}$ is an $\ell$-tuple of variables and ${\overline {x}}$ is an $n$-tuple of variables.

The (right) Ziegler spectrum, $\operatorname {Zg} _{R}$, of R is the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted by $\operatorname {pinj} _{R}$, and the topology has the sets

    $(\varphi /\psi )=\{N\in \operatorname {pinj} _{R}\mid \varphi (N)\supsetneq \psi (N)\cap \varphi (N)\}$

as subbasis of open sets, where $\varphi ,\psi$ range over (right) pp-1-formulae and $\varphi (N)$ denotes the subgroup of $N$ consisting of all elements that satisfy the one-variable formula $\varphi$. One can show that these sets form a basis.

## Properties

Ziegler spectra are rarely Hausdorff and often fail to have the $T_{0}$-property. However they are always compact and have a basis of compact open sets given by the sets $(\varphi /\psi )$ where $\varphi ,\psi$ are pp-1-formulae.

When the ring R is countable $\operatorname {Zg} _{R}$ is sober. It is not currently known if all Ziegler spectra are sober.

## Generalization

Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherent Grothendieck category, which generalizes the construction above.

