A stupid question: which statements in section 5 of BBD will fail if we replace $\overline{\mathbb{Q}_l}$-sheaves by just $\mathbb{Q}_l$-ones? I am especially interested in Proposition 5.1.15.
BBD = Beilinson A., Bernstein J., Deligne P., Faisceaux pervers, Asterisque 100, 1982, 5-171.
I think that all the statements are true, except for 5.3.9 (ii). Remark 5.3.10 says that all the statements in 5 up to and including 5.3.8 are true for $\mathbb{Q}_\ell$-coefficients with the same proof, and that 5.3.9 (i) is still true but with a slightly more complicated proof. I am pretty sure that the proof of corollary 5.3.10 works too for $\mathbb{Q}_\ell$-coefficients. I cannot see any reason why the proofs in 5.4 would not also work, but I have read them quickly. Anyway, proposition 5.1.15 is certainly fine.
Most of the statements go through. This is discussed in some detail at the start of Chapter 4. They also give a list of things that one has to be careful of when using other coefficients.
In particular, I think Proposition 5.1.15 stays true with coefficients in $\mathbb{Q}_{\ell}$. This should follow because all functors commute with the extension of scalars functor from $\mathbb{Q}_{\ell}$ to $\overline{\mathbb{Q}}_{\ell}$.
