The Sebastiani-thom Isomorphism in the Derived Category

Let f : X → C and g : Y → C be analytic functions. Let π1 and π2 denote the projections of X×Y onto X and Y , respectively. In [S-T], Sebastiani and Thom prove that the cohomology of the Milnor fibre of f ◦ π1 + g ◦ π2 is isomorphic to the tensor product of the cohomologies of the Milnor fibres of f and g (with a shift in degrees); they prove this in the case where X and Y are smooth and f and g have isolated critical points. In addition, they prove that the monodromy isomorphism induced by f ◦ π1 + g ◦ π2 is the tensor product of those induced by f and g. The point, of course, is to break up the complicated critical activity of f ◦ π1 + g ◦ π2 into more manageable pieces. Sebastiani-Thom-type results have been proved by Némethi [N1], [N2], Oka [O], and Sakamoto [S].