  We extend work of Balmer, associating filtrations of essentially small tensor
triangulated categories to certain dimension functions, to the setting of
actions of rigidly-compactly generated tensor triangulated categories on
compactly generated triangulated categories. We show that the towers of
triangles associated to such a filtration can be used to produce filtrations of
Gorenstein injective quasi-coherent sheaves on Gorenstein schemes. This extends
and gives a new proof of a result of Enochs and Huang. In the case of local
complete intersections a further refinement of this filtration is given and we
comment on some special properties of the associated spectral sequence in this
case.
