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Ziegler spectrum

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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

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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

where are natural numbers, is an matrix with entries from R, and is an -tuple of variables and is an -tuple of variables.

The (right) Ziegler spectrum, , of R is the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted by , and the topology has the sets

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

Properties

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Ziegler spectra are rarely Hausdorff and often fail to have the -property. However they are always compact and have a basis of compact open sets given by the sets where are pp-1-formulae.

When the ring R is countable is sober.[2] It is not currently known if all Ziegler spectra are sober.

Generalization

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Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherent Grothendieck category, which generalizes the construction above.[3]

References

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  1. ^ Ziegler, Martin (1984-04-01). "Model theory of modules" (PDF). Annals of Pure and Applied Logic. SPECIAL ISSUE. 26 (2): 149–213. doi:10.1016/0168-0072(84)90014-9.
  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. 74 (3): 503–558. doi:10.1112/S002461159700018X.