Question

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.

Answer 1

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.

Answer 2

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 $\overline{\mathbb{Q}}_{\ell}$.