When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

Question

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).

For $i,j\in \mathbb{Z}$ one can define the corresponding "naive" cohomology group as $\varprojlim_n H^i(F, \mathbb{Z}/ l^n\mathbb{Z}(j))$. This is not a very "good" definition; the "correct" cohomology groups of $F$ are the continuous ones (defined by Jannsen) that take into account the (first) derived projective limit (functor) for the system $H^i(F, \mathbb{Z}/ l^n\mathbb{Z}(j))$. My question is: when does the latter $\varprojlim{}^1$-group necessarily vanishes or (at least) torsion? It seems that the Bloch-Kato conjecture yields: the transition maps are surjective (and so, $\varprojlim{}^1$ vanishes) if $j=i$. Is this correct? Are there any general results of this sort available for $j\neq i$ (and if $K$ does not contain all $\mu_{l^n}$)? For example, what is known for $j=1$, $i=2$ (this case is closely related to the Brauer group)?

Answer 1

There are two questions here: when does the $\varprojlim\nolimits^1$ of a sequence of abelian groups $M_1 \longleftarrow M_2\longleftarrow M_3\dotsb$ vanish, and what can be said about the Galois cohomology with cyclotomic coefficients.

Concerning the first question: this is called the Mittag-Leffler condition, and I recall it being formulated as follows. The $\varprojlim\nolimits^1$ vanishes if, and perhaps also only if, for every $i>0$ the decreasing sequence of subgroups $\operatorname{im}(M_i\leftarrow M_j)$ eventually stabilizes in $M_i$ as $j$ grows to infinity.

Concerning the second question: certainly, it follows from the Milnor-Bloch-Kato conjecture that $\varprojlim_n{}^1 H^i(F,\mathbb Z/l^n\mathbb Z(j))=0$ when $i=j$, because this is a sequence of surjective maps of abelian groups. For $i\ne j$, this $\varprojlim\nolimits^1$ vanishing should not be true in general.