Grothendieck Rings of Theories of Modules

Abstract

The model-theoretic Grothendieck ring of a first order structure, as defined by Krajicěk 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.

42 Pages