  In a wide class of D-dimensional spacetimes which are direct or semi-direct
sums of a (D-n)-dimensional space and an n-dimensional homogeneous ``internal''
space, a field can be decomposed into modes. As a result of this mode
decomposition, the main objects which characterize the free quantum field, such
as Green functions and heat kernels, can effectively be reduced to objects in a
(D-n)-dimensional spacetime with an external dilaton field. We study the
problem of the dimensional reduction of the effective action for such
spacetimes. While before renormalization the original D-dimensional effective
action can be presented as a ``sum over modes'' of (D-n)-dimensional effective
actions, this property is violated after renormalization. We calculate the
corresponding anomalous terms explicitly, illustrating the effect with some
simple examples.
