nLab Ziegler spectrum

Let $k$ be a fixed field. Consider associative $k$-algebra $A$ and its category of right modules $Mod_A$.

Recall that a monomorphism $f:M\to N$ in $Mod_A$-modules is pure if after tensoring with any left $A$-module $L$ gives a mono $M\otimes_A L\stackrel{f\otimes L}\to N\otimes_A L$. A module $M$ in $Mod_A$ is pure-injective if every pure mono $M\to N$ splits. This is clearly a weaker property than being an injective object.

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

The importance of Ziegler spectrum is in the

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

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 $A$-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