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The model completion of the theory of modules over finitely generated commutative algebras

Published online by Cambridge University Press:  12 March 2014

Moshe Kamensky
Moshe Kamensky*
Affiliation:
Department of Mathematics, The Hebrew University, Jerusalem, Israel
*
Department of Pure Math, University of Waterloo, Waterloo, N2L 3G1, Ontario, Canada, E-mail: mkamensky@math.uwaterloo.ca, URL: http://mkamensky.notlong.com
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Abstract

We find the model completion of the theory modules over , where is a finitely generated commutative algebra over a field K. This is done in a context where the field K and the module are represented by sorts in the theory, so that constructible sets associated with a module can be interpreted in this language. The language is expanded by additional sorts for the Grassmanians of all powers of Kn, which are necessary to achieve quantifier elimination.

The result turns out to be that the model completion is the theory of a certain class of “big” injective modules. In particular, it is shown that the class of injective modules is itself elementary. We also obtain an explicit description of the types in this theory.

Keywords

modulesmodel completionquantifier eliminationPrimary 03C10Secondary 03C60
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 734 - 750
Copyright
Copyright © Association for Symbolic Logic 2009

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