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.^[1]

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

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

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

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

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

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

Properties

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

When the ring R is countable ${\displaystyle \operatorname {Zg} _{R}}$ is sober.^[2] 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.^[3]