  In this paper, we prove a version of Freyd's generating hypothesis for
triangulated categories: if D is a cocomplete triangulated category and S is an
object in D whose endomorphism ring is graded commutative and concentrated in
degree zero, then S generates (in the sense of Freyd) the thick subcategory
determined by S if and only if the endomorphism ring of S is von Neumann
regular. As a corollary, we obtain that the generating hypothesis is true in
the derived category of a commutative ring R if and only if R 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.
