Ziegler spectrum

Let $k$ be a fixed field. Consider associative $k$-algebra $A$ and its category of right modules ${\mathrm{Mod}}_{A}$.

Recall that a monomorphism $f:M\to N$ in ${\mathrm{Mod}}_{A}$-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 ${\mathrm{Mod}}_{A}$ 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** $\mathrm{Zsp}(A)$ 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 ${\mathrm{Mod}}_{A}$ and closed subsets of $\mathrm{Zsp}(A)$.

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)