# Six operations passage from $X_0$ to $X$ reference request

Let $X_0$ be a variety over $\mathbb F_q$ and denote by $X$ its basechange to the algebraic closure. Consider the constructible derived categories $D^b_c(X_0,\mathbb E)$ and $D^b_c(X,\mathbb E)$, where $l$ does not divide $q$ and $\mathbb E=\mathbb F_l,\mathbb Z_l,\mathbb Q_l$.

We have a morphism $X \rightarrow X_0$ and pullback along it gives a functor $$v:D^b_c(X_0,\mathbb E) \rightarrow D^b_c(X,\mathbb E)$$ which is usually denoted by $\mathcal F_0 \mapsto \mathcal F$.

I would like to have a reference for the fact, that $v$ "commutes" with the six functors $$\cal Hom,\otimes, f_*,f_!,f^*,f^!$$ So for example I would like to know that $v(f_* \mathcal F_0) =f_* \mathcal F$ or $\mathcal vHom(\mathcal F_0,\mathcal G_0)=\mathcal Hom(\mathcal F,\mathcal G)$

• The six functors are treated in "Laszlo, Yves; Olsson, Martin, The six operations for sheaves on Artin stacks. II. Adic coefficients. Publ. Math. IHES No. 107 (2008), 169--210" and "Liu and Zheng, Enhance six operations and base change theorem, preprint May 2012; arXiv:1211:5948. I don't know whether they explicitly prove they commute with base change, but this should be obvious from the definitions. – abz Aug 13 '13 at 11:56
• For $Rf_!$ with torsion coefficients, this is SGA4 Exp XVII Theorem 5.2.6. – S. Carnahan Aug 13 '13 at 13:50
• Thanks! @anon The problem is, that the algebraic closure is not of finte type over the groundfield. In the sources you suggest, everything is assumed to be of finite type. – Jan Weidner Aug 14 '13 at 8:23
• All formulas would follow, if $X\rightarrow X_0$ was of finite type (and hence etale). Now $X\rightarrow X_0$ is at least a limit of etale maps, but I don't know how to exploit this. – Jan Weidner Aug 14 '13 at 10:03