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.

  • Martin Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149–213, MR86c:03034 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
Revised on May 23, 2012 17:18:34 by Zoran Škoda (193.51.104.33)