In a wide class of four-dimensional spacetimes which are direct or semidirect products of a homogeneousn-dimensional space and a(4−n)-dimensional 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(4−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 four-dimensional object can be presented as a “sum over modes” of(4−n)-dimensional objects, this property is violated after renormalization. We calculate the corresponding anomalous terms and relate their origin with the effect of the multiplicative anomaly. This effect is demonstrated with some simple examples. ©1999 American Institute of Physics.