The generating hypothesis in the derived category of \(R\)-modules
Freyd's generating hypothesis states that a map between finite spectra that induces the zero map on stable homotopy groups is null-homotopic [\textit{P. Freyd}, Proc. Conf. Categor. Algebra, La Jolla 1965, 121--172 (1966; Zbl 0195.52901)]. In this paper, a version of Freyd's generating hypothesis is considered for a triangulated category \(\mathcal D\) with a distinguished object \(S\) such that the endomorphism ring \(R\) of \(S\) is commutative and concentrated in degree zero. The main theorem is that if the category \(\mathcal D\) admits arbitrary coproducts, then the generating hypothesis is true if and only if \(R\) is von Neumann regular. Here, a commutative ring is von Neumann regular if every principal ideal is generated by an idempotent. An example of such category is the derived category of \(R\)-modules for commutative ring \(R\). The necessity of the theorem follows from two necessary conditions which are equivalent to the condition that \(R\) is von Neumann regular. The sufficiency is proved by showing a property of a von Neumann regular ring \(R\), namely that every finitely generated submodule of a projective \(R\)-module is a summand. Furthermore, similar results are obtained by replacing the finiteness condition, and in a Noetherian stable homotopy category.
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