Relative singularity category of a non-commutative resolution of   singularities
  In this article we study the triangulated category of singularities associated with a non-commutative resolution of singularities. In particular, we give a complete description of this category in the case of a curve with nodal singularities, classifying its indecomposable objects and computing its Auslander-Reiten quiver and K-group. 
Igor Burban and Martin Kalck
math.AG math.RT