A Thom-Sebastiani theorem in characteristic $p$

Let $k$ be a perfect field of characteristic $p$, $X_i$ $(i=1,2)$ smooth $k$-schemes, $f_i:X_i\to\mathbb A_k^1$ two $k$-morphisms of finite type, and $f:X_1\times_k X_2\to \mathbb A_k^1$ 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 cycles complex of $f$ at $x=(x_1,x_2)$ is the convolution product of the vanishing cycles complexes of $f_i$ at $x_i$ $(i=1,2)$.

Keywords

vanishing cycles, nearby cycles, local Fourier transformation, perverse sheaf

2010 Mathematics Subject Classification

14F20

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Published 25 July 2014