nLab
Ziegler spectrum

Let kk be a fixed field. Consider associative kk-algebra AA and its category of right modules Mod AMod_A.

Recall that a monomorphism f:MNf:M\to N in Mod AMod_A-modules is pure if after tensoring with any left AA-module LL gives a mono M ALfLN ALM\otimes_A L\stackrel{f\otimes L}\to N\otimes_A L. A module MM in Mod AMod_A is pure-injective if every pure mono MNM\to N splits. This is clearly a weaker property than being an injective object.

Given an associative algebra AA, its Ziegler spectrum Zsp(A)Zsp(A) is the topological space whose points are the isomorphism classes [M][M] of indecomposable pure-injective AA-modules MM and the topology is defined in terms of pp-formulas (or finite matrices) over AA. Here pp stands for “positive primitive in the usual language for AA-modules”

The importance of Ziegler spectrum is in the

Ziegler’s theorem. There is a correspondence between the definable classes in Mod AMod_A and closed subsets of Zsp(A)Zsp(A).

There are applications to the spectra of theories of modules.

It is introduced in

  • Martin Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149–213, MR86c:03034 doi

and generalized to locally coherent Grothendieck categories in

  • Ivo Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. 74 (3): 503–558 (1997) doi

  • Krause pdf

  • Grigory Garkusha, Mike Prest (2005) Triangulated categories and the Ziegler spectrum, Algebras and Representation Theory, 8 (4). pp. 499-523, doi, pdf

  • Mike Prest, slides

  • Mike Prest, Topological and geometric aspects of the Ziegler spectrum (1998)

  • Lorna Gregory, Thesis, pdf

  • Mike Prest, Purity, spectra and localisation, Enc. of Math. and its Appl. 121, Camb. Univ. Press 2009

On the relation between the Ziegler’s spectrum of the category of finitely generated left AA-modules and the Gabriel’s spectrum of the category of Abelian presheaves on it see

  • Ivo Herzog, Contravariant functors on the category of finitely presented modules, Israel J. Math. 167, 347–410 (2008) doi

Last revised on January 22, 2021 at 13:00:56. See the history of this page for a list of all contributions to it.