Picard–Lefschetz monodromy of products

Let ff and gg be reduced homogeneous polynomials in separate sets of variables. We establish a simple formula that relates the eigenspace decomposition of the monodromy operator on the Milnor fiber cohomology of fgfg to that of ff and gg separately. We use a relation between local systems and Milnor fiber cohomology that has been established by D. Cohen and A. Suciu.