A Thom-Sebastiani Theorem in Characteristic p

Abstract

Let $k$ be a perfect field of characteristic $p$, let $f_i:X_i\to {\mathbb{A}}_k^1$ $(i=1,2)$ be two $k$-morphism of finite type, and let $f:X_1{\times }_k{X}_2\to {\mathbb{A}}_k^1$ be the morphism defined by $f(z_1,z_2)=f_1(z_1)+f_2(z_2)$. For each $i\in \{1,2\}$, let $x_i$ be a $k$-rational point in the fiber $f_i^{-1}(0)$ such that $f_i$ is smooth on $X_i-\{x_i\}$. Using the $\ell$-adic Fourier transformation and the stationary phase principle of Laumon, we prove that the vanishing cycle of $f$ at $x=(x_1,x_2)$ is the convolution product of the vanishing cycles of $f_i$ at $x_i$ $(i=1,2)$.

arXiv commentFinal version. To appear in Mathematical Research Letters

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