Given a finitely generated field $F$ with prime field $k$, we assume $k$ is finite, of characteristic $p$. Fix a prime $\ell$ invertible in $k$.

In the discussion right after [K, Lemma 2.3], the variant

$$H^{*,\star}:=\tilde{\rm H}^*_{\rm cont}(\text{Spec}(F)/\text{Spec}(k),\mathbf{Z}_{\ell}(\star))$$ of Jannsen's continuous étale cohomology of $\text{Spec}(F)$ is defined, for integers $(*,\star)$, with $*$ nonnegative (notation as in loc cit, with $X = \text{Spec}(F)$ and $S = \text{Spec}(k)$).

A more explicit description of $H^{*,\star}$ can be given, as per loc cit, as follows.

Write $F$ as a filtered colimit of the global complete intersection $k$-subalgebras $A$ of $F$ (see [SP, [Tag 07BV](https://stacks.math.columbia.edu/tag/07BV))], and denote by ${\rm H}_{\rm cont}^*(\text{Spec}(A),\mathbf{Z}_{\ell}(\star))$ the continuous étale cohomology of Jannsen's. 

**Remark.** Here $\mathbf{Z}_{\ell}(\star)$ is the pro-sheaf $\{\mu_{\ell^n}^{\otimes\star}, n\ge 1\}$, as per construction of Jannsen's continuous étale cohomology. See [J] for details on continuous étale cohomology, and [K, $\S$2] for an overview. Roughly, continuous étale cohomology of a scheme $X$ with coefficients in the pro-sheaf $\mathbf{Z}_{\ell}(\star)$ is, degree-wise, an extension of usual $\ell$-adic cohomology of $X$ by a $\lim^1$ module.

Then the $H^{*,\star}$ from above can be written as
$$H^{*,\star} = \varinjlim_{k\subset A\subset F}{\rm H}_{\rm cont}^*(\text{Spec}(A),\mathbf{Z}_{\ell}(\star)).$$

Now define
$$H^{*,\star}_{\mathbf{Q}} := H^{*,\star}\otimes_{\mathbf{Z}}\mathbf{Q}.$$

> **Question.**  Why does $H^{i,p}_{\mathbf{Q}}$ vanish for $i>p+1$, regardless of the transcendence degree of $F$?

**Example.** When $F$ is a *finite* extension of $k$, then we get
$$H^{*,\star} = {\rm H}^*_{\rm cont}(F,\mathbf{Z}_{\ell}(\star)).$$
Here the right side is continuous cohomology of the field $F$ as in [J, Thm. 3.2]. Since $F$, in this example, is finite, we get that, for all integers $p$, $H^{i,p}$ is zero for all $i\neq 1$, and finite for $i=1$, so $H^{i,p}_{\mathbf{Q}}=0$ for all $i$.

**References:**

[J] U. Jannsen, *Continuous étale cohomology*. Math. Annalen.\
[K] B. Kahn, *A sheaf-theoretic  reformulation of the Tate conjecture*. [Preprint](https://arxiv.org/pdf/math/9801017.pdf).\
[SP] [Stacks Project](https://stacks.math.columbia.edu/tag/07BV).