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
- [math]\displaystyle{ \exists \overline{y} \ (\overline{y},\overline{x}) A=0 }[/math]
where [math]\displaystyle{ \ell,n,m }[/math] are natural numbers, [math]\displaystyle{ A }[/math] is an [math]\displaystyle{ (\ell+n)\times m }[/math] matrix with entries from R, and [math]\displaystyle{ \overline{y} }[/math] is an [math]\displaystyle{ \ell }[/math]-tuple of variables and [math]\displaystyle{ \overline{x} }[/math] is an [math]\displaystyle{ n }[/math]-tuple of variables.
The (right) Ziegler spectrum, [math]\displaystyle{ \operatorname{Zg}_R }[/math], of R is the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted by [math]\displaystyle{ \operatorname{pinj}_R }[/math], and the topology has the sets
- [math]\displaystyle{ (\varphi/\psi) = \{N\in\operatorname{pinj}_R \mid \varphi(N) \supsetneq \psi(N)\cap\varphi(N)\} }[/math]
as subbasis of open sets, where [math]\displaystyle{ \varphi,\psi }[/math] range over (right) pp-1-formulae and [math]\displaystyle{ \varphi(N) }[/math] denotes the subgroup of [math]\displaystyle{ N }[/math] consisting of all elements that satisfy the one-variable formula [math]\displaystyle{ \varphi }[/math]. One can show that these sets form a basis.
Properties
Ziegler spectra are rarely Hausdorff and often fail to have the [math]\displaystyle{ T 0 }[/math]-property. However they are always compact and have a basis of compact open sets given by the sets [math]\displaystyle{ (\varphi/\psi) }[/math] where [math]\displaystyle{ \varphi,\psi }[/math] are pp-1-formulae.
When the ring R is countable [math]\displaystyle{ \operatorname{Zg}_R }[/math] 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]
References
- ↑ Ziegler, Martin (1984-04-01). "Model theory of modules" (in en). Annals of Pure and Applied Logic. SPECIAL ISSUE 26 (2): 149–213. doi:10.1016/0168-0072(84)90014-9. http://home.mathematik.uni-freiburg.de/ziegler/preprints/modules/modules.pdf.
- ↑ Ivo Herzog (1993). Elementary duality of modules. Trans. Amer. Math. Soc., 340:1 37–69
- ↑ Herzog, I. (1997). "The Ziegler Spectrum of a Locally Coherent Grothendieck Category" (in en). Proceedings of the London Mathematical Society 74 (3): 503–558. doi:10.1112/S002461159700018X. https://www.researchgate.net/publication/231890405.
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