The generating hypothesis in the derived category of a ring
Mark Hovey, Keir Lockridge, and Gena Puninski
We show that a strong form (the fully faithful version) of the
generating hypothesis, introduced by Freyd in algebraic topology, holds
in the derived category of a ring $R$ if and only if $R$ is von Neumann
regular. This extends results of the second author~\cite{lockridge}.
We also characterize rings for which the original form (the faithful
version) of the generating hypothesis holds in the derived category of
$R$. These must be close to von Neumann regular in a precise sense,
and, given any of a number of finiteness hypotheses, must be von Neumann
regular. However, we construct an example of such a ring that is not
von Neumann regular and therefore does not satisfy the strong form of
the generating hypothesis.