  In this paper, we construct a version of Auslander-Reiten sequences for the
$K(n)$-local stable homotopy category. In particular, the role of the
Auslander-Reiten translation is played by the local Brown-Comenetz duality
functor. As an application, we produce counterexamples to the $K(n)$-local
generating hypothesis for all heights $n>0$ and all primes. Furthermore, our
methods apply to other triangulated categories, as for example the derived
category of quasi-coherent sheaves on a smooth projective scheme.
