  In this short paper we introduce a new triangulated category for rational
surface singularities which in the non-Gorenstein case acts as a substitute for
the stable category of matrix factorizations. The category is formed as a
Frobenius quotient of the category of special CM modules, and we classify the
relatively projective-injective objects and thus describe the AR quiver of the
quotient. Connections to the corresponding reconstruction algebras are also
discussed.
