Wildly Compatible Systems and Six Operations

Abstract

For a scheme $X$ separated and of finite type over an excellent regular scheme $S$, we define wildly compatible systems of constructible sheaves of modules over finite fields on $X$ for certain vector spaces $V$. The main result is that for $\dim S \leq 1$, wildly compatible systems are preserved by Grothendieck's six operations and Verdier's duality. Finally, for a smooth integral scheme $X$ over a finite field, we prove that all $\ell$-adic compatible systems gives wildly compatible systems.