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Let $X_0$ be a variety over $\mathbb F_q$. Denote by $X_n$ its basechange to $\mathbb F_{q^n}$ and let $X=\lim X_n$ be its basechange to the algebraic closure $\overline{\mathbb F}_q$.

Let $D^b_c(X_n,\mathbb F_l)$ be the constructible dervied category with $\mathbb F_l$ coefficients. Is it true, that the map

$$colim D^b_c(X_n,\mathbb F_l)\rightarrow D^b_c(X,\mathbb F_l)$$

is an equivalence of categories?

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  • $\begingroup$ Shouldn't you start with $X_1$? Also, is your (2-)colimit defined by a compatible system of $\ast$-pullbacks? $\endgroup$
    – S. Carnahan ♦
    Commented Aug 28, 2013 at 12:30
  • $\begingroup$ Yes, the 2-colimit is over $*$-pullbacks.You are right, the notation $X_1$ would be the more logical for what I call $X_0$. On the other hand the notation $X_0$ seems to be standard. $\endgroup$
    – Jan Weidner
    Commented Aug 28, 2013 at 12:48

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