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The torsionfree part of the Ziegler spectrum of RG when R is a Dedekind domain and G is a finite group

Published online by Cambridge University Press:  12 March 2014

A. Marcja ,
M. Prest  and
C. Toffalori
A. Marcja
Affiliation:
Dipartimento Di Matematica, U. Dini, Università Di Firenze, Viale Morgagni 67/A, 1-50134 Firenze, Italy, E-mail: [email protected]
M. Prest
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England, E-mail: [email protected]
C. Toffalori
Affiliation:
Dipartimento Di Matematica E Infomatica, Università Di Camerino, Via Madonna Delle Carceri, 1-62032 Camerino, Italy, E-mail: [email protected]
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Extract

For every ring S with identity, the (right) Ziegler spectrum of S, Zgs, is the set of (isomorphism classes of) indecomposable pure injective (right) S-modules. The Ziegler topology equips Zgs with the structure of a topological space. A typical basic open set in this topology is of the form

where φ and ψ are pp-formulas (with at most one free variable) in the first order language Ls for S-modules; let [φ/ψ] denote the closed set Zgs - (φ/ψ). There is an alternative way to introduce the Ziegler topology on Zgs. For every choice of two f.p. (finitely presented) S-modules A, B and an S-module homomorphism f: A → B, consider the set (f) of the points N in Zgs such that some S-homomorphism h: A → N does not factor through f. Take (f) as a basic open set. The resulting topology on Zgs is, again, the Ziegler topology.

The algebraic and model-theoretic relevance of the Ziegler topology is discussed in [Z], [P] and in many subsequent papers, including [P1], [P2] and [P3], for instance. Here we are interested in the Ziegler spectrum ZgRG of a group ring RG, where R is a Dedekind domain of characteristic 0 (for example R could be the ring Z of integers) and G is a finite group. In particular we deal with the R-torsionfree points of ZgRG.

The main motivation for this is the study of RG-lattices (i.e., finitely generated R-torsionfree RG-modules).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 3 , September 2002 , pp. 1126 - 1140
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[BT]Baratella, S. and Toffalori, C., The theory of ZC (2)2-lattices is decidable, Archive of Mathematical Logic, vol. 37 (1998), pp. 91–104.CrossRefGoogle Scholar
[CR]Curtis, C. and Reiner, I., Methods of representation theory with applications to finite groups and orders I, Wiley, New York, 1981.Google Scholar
[J]Jacobson, N., Basic algebra II, W. H. Freeman, San Francisco, 1980.Google Scholar
[MT]Marcja, A. and Toffalori, C., Decidable representations, Journal of Pure and Applied Algebra, vol. 103 (1995), pp. 189–203.CrossRefGoogle Scholar
[P]Prest, M., Model theory and modules, Cambridge University Press, Cambridge, 1988.CrossRefGoogle Scholar
[P2]Prest, M., The Zariski spectrum of the category of finitely presented modules, preprint, 1998.Google Scholar
[P1]Prest, M., Ziegler spectra of tame hereditary algebras, Journal of Algebra, vol. 207 (1998), pp. 146–164.CrossRefGoogle Scholar
[P3]Prest, M., Topological and geometric aspects of the Ziegler spectrum, Infinite length modules (Krause, H. and Ringel, C. M., editors), Birkhäuser, 2000, pp. 369–392.CrossRefGoogle Scholar
[Roth]Rothmaler, Ph., Purity in model theory, Advances in algebra and model theory, Algebra, Logic and Applications, vol. 9, Gordon and Breach, Amsterdam, 1997, pp. 445–469.Google Scholar
[T]Toffalori, C., Wildness implies undecidability for lattices over group rings, this Journal, vol. 63 (1997), pp. 1429–1447.Google Scholar
[T1]Toffalori, C., The decision problem for ZC(p3)-lattices with p prime, Archive of Mathematical Logic, vol. 37 (1998), pp. 127–142.CrossRefGoogle Scholar
[Z]Ziegler, M., Model theory of modules, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 149–213.CrossRefGoogle Scholar