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Model theory of comodules

Published online by Cambridge University Press:  12 March 2014

Septimiu Crivei
Affiliation:
Department of Mathematics, Babes-Bolyai University, Cluj, Romania, E-mail: crivei@math.ubbcluj.ro
Mike Prest
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester. M13 9PL., Great Britain, E-mail: mprest@maths.man.ac.uk
Geert Reynders
Affiliation:
Kapellestraat 10, 3080 Tervuren., Belgium, E-mail: geert.reyndersl@pandora.be

Extract

The purpose of this paper is to establish some basic points in the model theory of comodules over a coalgebra. It is not even immediately apparent that there is a model theory of comodules since these are not structures in the usual sense of model theory. Let us give the definitions right away so that the reader can see what we mean.

Fix a field k. A k-coalgebra C is a k-vector space equipped with a k-linear map Δ: CCC, called the comultiplication (by ⊗ we always mean tensor product over k), and a k-linear map ε: Ck, called the counit, such that Δ⊗1C = 1C ⊗ Δ (coassociativity) and (1Cε)Δ = 1C = (ε ⊗ 1C)Δ, where we identify C with both kC and Ck. These definitions are literally the duals of those for a k-algebra: express the axioms for C′ to be a k-algebra in terms of the multiplication map μ: C′ ⊗ C′ → C′ and the “unit” (embedding of k into C′), δ: kC′ in the form that certain diagrams commute and then just turn round all the arrows. See [5] or more recent references such as [7] for more.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2004

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References

[1]Adámek, J. and Rosický, J., Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, 1994.CrossRefGoogle Scholar
[2]Crawley-Boevey, W., Locally finitely presented additive categories, Communications in Algebra, vol. 22 (1994), pp. 16411674.CrossRefGoogle Scholar
[3]Prest, M. and Wisbauer, R., Finite presentation and purity in categories σ[M], Colloquium Mathematicum, to appear.Google Scholar
[4]Reynders, G.. Ziegler spectra over serial rings and coalgebras, Doctoral thesis. University of Manchester, 1998.Google Scholar
[5]Sweedler, M. E., Hopf algebras, W. A. Benjamin, 1969.Google Scholar
[6]Wisbauer, R., Foundations of module and ring theory, Gordon and Breach, Philadelphia, 1991.Google Scholar
[7]Wisbauer, R., Module and comodule categories—A survey, Proceedings of the mathematics conference (Birzeit University 1998), World Scientific, 2000, pp. 277304.Google Scholar

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