2 $\mathbb{Z}[1/l]$ instead of $\mathbb{Z}_l$; deleted 2 characters in body
# Is it easy to define weights for $Q_l$-sheaves over finite type $Z_l$-schemes?
In her paper "Mixed perverse sheaves for schemes over number fields" (http://www.springerlink.com/content/w848j73552102274/) A. Huber defines certain weights for certain categories of $\mathbb{Q}_l$-sheaves over a finite type $\mathbb{Q}$-scheme $S$ using the following argument:
she presents $S$ as a limit of finite type $U_0$-schemes $S_0$, where $U_0$ is an open subscheme of $\mathbb{Z}_l$. Then (as far as I understand) she defines her (derived) category of sheaves over $S$ as a 2-limit of those for all $S_0$. She states that there exist a perverse $t$-structure and the nice notion of weights for this limit category that could be obtained as the limit of the corresponding structures over all $S_0$. So, it seems that the corresponding properties of sheaves and weights over $S_0$ (see Huber's 3.4-3.9) should be true (and the proof is even easier; Huber just relies on the corresponding results of BBD). Is this correct, or is there some matter that simplifies when we pass from $S_0$ to $S$?