Orlovs famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this result is false without the full faithfulness hypothesis.
In the recent paper Mutation in triangulated categories and rigid Cohen-Macaulay modules Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay mod
ules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlovs result on the graded singularity category. We obtain some new results on the singularity category of isolated singularities which may be interesting in their own right.
