It is said that renormalization in curved spacetime is difficult. But technically, renormalization procedures can be translated into a problem adding counterterms into Lagrangian. Can't this Lagrangian, translated to accommodate general spacetime metric, be used to define renormalization procedure?
I could understand that even if the above procedure is valid, since we do not in general know how to decompose solution to the equation of motion, we really do not know what we are really getting out of this procedure.
But my question is restricted to whether such Lagrangian procedure would technically be the valid approach.