The generating hypothesis in the derived category of RR-modules

In this paper, we prove a version of Freyd’s generating hypothesis for triangulated categories: if DD is a cocomplete triangulated category and S∈DS∈D is an object whose endomorphism ring is graded commutative and concentrated in degree zero, then SS generates (in the sense of Freyd) the thick subcategory determined by SS if and only if the endomorphism ring of SS is von Neumann regular. As a corollary, we obtain that the generating hypothesis is true in the derived category of a commutative ring RR if and only if RR is von Neumann regular. We also investigate alternative formulations of the generating hypothesis in the derived category. Finally, we give a characterization of the Noetherian stable homotopy categories in which the generating hypothesis is true.