Ziegler spectrum explained

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

$$\backslash exists\; \backslash overline\; \backslash \; (\backslash overline,\backslash overline)\; A=0$$

where

\ell,n,m are natural numbers, A is an (\ell + n) x 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 topologyhas the sets

$$(\backslash varphi/\backslash psi)\; =\; \backslash$$

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.^[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]

Notes and References

1. Ziegler. Martin. 1984-04-01. Model theory of modules. Annals of Pure and Applied Logic. SPECIAL ISSUE. en. 26. 2. 149–213. 10.1016/0168-0072(84)90014-9. free.
2. Ivo Herzog (1993). Elementary duality of modules. Trans. Amer. Math. Soc., 340:1 37–69
3. Herzog. I.. 1997. The Ziegler Spectrum of a Locally Coherent Grothendieck Category. Proceedings of the London Mathematical Society. en. 74. 3. 503–558. 10.1112/S002461159700018X.