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 "intern
al" space, a field can be decomposed into modes. As a result of this mode d
ecomposition, the main objects which characterize the free quantum field, s
uch as Green functions and heat kernels, can effectively be reduced to obje
cts in a (D-n)-dimensional spacetime with an external dilaton field. We stu
dy the problem of the dimensional reduction of the effective action for suc
h spacetimes. While before renormalization the original D-dimensional effec
tive action can be presented as a "sum over modes'' of (D-n)-dimensional ef
fective actions, this property is violated after renormalization. We calcul
ate the corresponding anomalous terms explicitly, illustrating the effect w
ith some simple examples.