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
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
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