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arXiv · 1302.4229

Grothendieck Rings of Theories of Modules

Amit Kuber
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Abstract

The model-theoretic Grothendieck ring of a first order structure, as defined by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, $K_0(M_\mathcal R)$, of a right $R$-module $M$, where $\mathcal R$ is any unital ring. As a corollary we prove a conjecture of Prest that $K_0(M)$ is non-trivial, whenever $M$ is non-zero. The main proof uses various techniques from the homology theory of simplicial complexes.

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Amit Kuber. 2013-02-18. Grothendieck Rings of Theories of Modules. https://arxiv.org/abs/1302.4229

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