Theorem 5.1 on page 80 of this paper says that

*Assuming that the matter fields satisfy their equations of motion, the matter field action is locally Weyl invariant if and only if the corresponding energy-momentum tensor is traceless.*

If I have understood this right, Weyl invariance reduces to scale invariance in flat spacetime, which is basically what I am up to at the moment.

My question now is:

If one has Weyl or scale invariance in accordance with this theorem, does this mean that global Weyl or scale invariance (such that the scaling parameter does not depend on position) is guaranteed too? If not why not?