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

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

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

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

$$(\phi /\psi )=\{N\in {\mathrm{pinj}}_R\mid \phi (N)⊋\psi (N)\cap \phi (N)\}$$

as subbasis of open sets, where $\phi ,\psi$ range over (right) pp-1-formulae and $\phi (N)$ denotes the subgroup of $N$ consisting of all elements that satisfy the one-variable formula $\phi$. 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 $(\phi /\psi )$ where $\phi ,\psi$ are pp-1-formulae.

When the ring R is countable ${\mathrm{Zg}}_R$ 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.