Around the Thom-Sebastiani theorem

TL;DR

This paper extends the classical Thom-Sebastiani theorem to algebraic and étale cohomology settings over fields of any characteristic, introducing a local convolution product replacing the tensor product and proving a K"unneth formula for nearby cycles.

Contribution

It provides algebraic variants and generalizations of the Thom-Sebastiani theorem using étale cohomology and local convolution, broadening its applicability across different characteristics.

Findings

Established a K"unneth formula for nearby cycles in Deligne's theory.

Generalized Thom-Sebastiani theorem to étale cohomology with local convolution.

Analyzed the relations between tensor and convolution products in tame cases.

Abstract

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 an isolated critical point at 0 with value 0, the classical Thom-Sebastiani theorem describes the vanishing cycles group $\Phi^{m+n+1}(f \oplus g)$ (and its 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 prove algebraic variants and generalizations of this result in \'etale cohomology over fields of any characteristic, where the tensor product is replaced by a certain local convolution product, as suggested by Deligne. They generalize arXiv:1105.5210. The main ingredient is a K\"unneth formula for $R\Psi$ in the framework of Deligne's theory of nearby cycles over general bases. In the last section, we study the tame case, and the relations between tensor and convolution products, in both global and local situations.