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For germs of holomorphic functions $f : (\mathbf{C}^{m+1},0) \to(\mathbf{C},0)$, $g : (\mathbf{C}^{n+1},0) \to (\mathbf{C},0)$ having anisolated critical point at 0 with value 0, the classical Thom-Sebastianitheorem describes the vanishing cycles group $\Phi^{m+n+1}(f \oplus g)$ (andits monodromy) as a tensor product $\Phi^m(f) \otimes \Phi^n(g)$, where $(f\oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n)$. We provealgebraic variants and generalizations of this result in \'etale cohomologyover fields of any characteristic, where the tensor product is replaced by acertain local convolution product, as suggested by Deligne. They generalizearXiv:1105.5210. The main ingredient is a K\"unneth formula for $R\Psi$ in theframework of Deligne's theory of nearby cycles over general bases. In the lastsection, we study the tame case, and the relations between tensor andconvolution products, in both global and local situations.

Around the Thom-Sebastiani theorem