Let $k$ be a field, $X$ a smooth projective variety over $k$, $\overline{X} := X\times_k {k}^{\rm sep}$ for a separable closure ${k}^{\rm sep}$ of $k$, $\ell$ a prime with $\ell\in k^{\times}$.

Are the Galois cohomology groups $$H^i(\text{Gal}({k}^{\rm sep}/k),H^j_{\rm ét}(\overline{X},\mathbf{Z}_{\ell}))$$ finite for $i\ge 1$, $j\ge 0$?

I would appreciate to get some references.

I have seen stated that $H^1(\text{Gal}({k}^{\rm sep}/k),H^j_{\rm ét}(\overline{X},\mathbf{Z}_{\ell}))$ is finite for $j\ge 0$ and $k$ a finite field, but I had the impression this should be a general fact about cohomology of profinite groups with coefficients in finitely generated $\mathbf{Z}_{\ell}$-modules with appropriate action.

Let $k$ be a field, $X$ a smooth projective variety over $k$, $\overline{X} := X\times_k {k}^{\rm sep}$ for a separable closure ${k}^{\rm sep}$ of $k$, $\ell$ a prime with $\ell\in k^{\times}$.

Are the Galois cohomology groups $$H^i(\text{Gal}({k}^{\rm sep}/k),H^j_{\rm ét}(\overline{X},\mathbf{Z}_{\ell}))$$ finite for $i\ge 1$, $j\ge 0$? Are they torsion?

I would appreciate to get some references.

I have seen stated that $H^1(\text{Gal}({k}^{\rm sep}/k),H^j_{\rm ét}(\overline{X},\mathbf{Z}_{\ell}))$ is finite for $j\ge 0$ and $k$ a finite field, but I had the impression this should be a general fact about cohomology of profinite groups with coefficients in finitely generated $\mathbf{Z}_{\ell}$-modules with appropriate action.

Let $k$ be a field, $X$ a smooth projective variety over $k$, $\overline{X} := X\times_k {k}^{\rm sep}$ for a separable closure ${k}^{\rm sep}$ of $k$, $\ell$ a prime with $\ell\in k^{\times}$.

Are the Galois cohomology groups $$H^i(\text{Gal}({k}^{\rm sep}/k),H^j_{\rm ét}(\overline{X},\mathbf{Z}_{\ell}))$$ finite for $i\ge 1$, $j\ge 0$?

I would appreciate to get some references. Thanks

Let $k$ be a field, $X$ a smooth projective variety over $k$, $\overline{X} := X\times_k {k}^{\rm sep}$ for a separable closure ${k}^{\rm sep}$ of $k$, $\ell$ a prime with $\ell\in k^{\times}$.

Are the Galois cohomology groups $$H^i(\text{Gal}({k}^{\rm sep}/k),H^j_{\rm ét}(\overline{X},\mathbf{Z}_{\ell}))$$ finite for $i\ge 1$, $j\ge 0$?

I would appreciate to get some references.

I have seen stated that $H^1(\text{Gal}({k}^{\rm sep}/k),H^j_{\rm ét}(\overline{X},\mathbf{Z}_{\ell}))$ is finite for $j\ge 0$ and $k$ a finite field, but I had the impression this should be a general fact about cohomology of profinite groups with coefficients in finitely generated $\mathbf{Z}_{\ell}$-modules with appropriate action.