If $f:X\to\text{Spec}(k)$ is a smooth projective variety over a separably closed field, with $i:Z\subset X$ a closed smooth sub variety, with complement $j: U\to X$, and $f':X'\to \text{Spec}(k)$ is the blow-up of $X$ along $Z$, we have, in $\ell$-adic cohomology:

$$(*)\ \ H^a(X',\mathbf{Z}_{\ell}(n)) = H^a(X,\mathbf{Z}_{\ell}(n))\oplus\bigoplus_{j=1}^{c-1}H^{a-2j}(Z,\mathbf{Z}_{\ell}(n-j)).$$

Is there, in addition, a splitting of the complex $Rf'_*\mathbf{Z}_{\ell}(n)$ in the derived category of $\ell$-adic sheaves? What is this splitting?

It should be something like

$$Rf'_*\mathbf{Z}_{\ell}(n) = \bigoplus_j \left(H^j(X,\mathbf{Z}_{\ell}(n))[-2j]\oplus i_*H^j(Z,\mathbf{Z}_{\ell}(n-c))[-2j+2c]\right)\ ?$$

I'd like to ask for a reference for the proof of the derived version of formula $(*)$, or an argument, with correct weights and shifts.

Thanks a lot!

If $f:X\to\text{Spec}(k)$ is a smooth projective variety over a separably closed field, with $i:Z\subset X$ a closed smooth sub variety, with complement $j: U\to X$, and $f':X'\to \text{Spec}(k)$ is the blow-up of $X$ along $Z$, we have, in $\ell$-adic cohomology:

$$(*)\ \ H^a(X',\mathbf{Z}_{\ell}(n)) = H^a(X,\mathbf{Z}_{\ell}(n))\oplus\bigoplus_{j=1}^{c-1}H^{a-2j}(Z,\mathbf{Z}_{\ell}(n-j)).$$

Is there, in addition, a splitting of the complex $Rf'_*\mathbf{Z}_{\ell}(n)$ in the derived category of $\ell$-adic sheaves? What is this splitting?

It should be something like

$$Rf'_*\mathbf{Z}_{\ell}(n) = Rf_*\mathbf{Z}_{\ell}(n)\oplus\bigoplus_j R^j(f\circ i)_*\mathbf{Z}_{\ell}(n-c)[-2j+2c]\ ?$$

I'd like to ask for a reference for the proof of the derived version of formula $(*)$, or an argument, with correct weights and shifts.

Thanks a lot!

If $f:X\to\text{Spec}(k)$ is a smooth projective variety over a separably closed field, with $i:Z\subset X$ a closed smooth sub variety, with complement $j: U\to X$, and $f':X'\to \text{Spec}(k)$ is the blow-up of $X$ along $Z$, we have, in $\ell$-adic cohomology:

$$(*)\ \ H^a(X',\mathbf{Z}_{\ell}(n)) = H^a(X,\mathbf{Z}_{\ell}(n))\oplus\bigoplus_{j=1}^{c-1}H^{a-2j}(Z,\mathbf{Z}_{\ell}(n-j)).$$

Is there, in addition, a splitting of the complex $Rf'_*\mathbf{Z}_{\ell}(n)$ in the derived category of $\ell$-adic sheaves? What is this splitting?

It should be something like

$$Rf'_*\mathbf{Z}_{\ell}(n) = Rf_*\mathbf{Z}_{\ell}(n)\oplus R(f\circ i)_*\mathbf{Z}_{\ell}(n-c)[2c]\ ?$$

I'd like to ask for a reference for the proof of the derived version of formula $(*)$, or an argument, with correct weights and shifts.

Thanks a lot!

If $f:X\to\text{Spec}(k)$ is a smooth projective variety over a separably closed field, with $i:Z\subset X$ a closed smooth sub variety, with complement $j: U\to X$, and $f':X'\to \text{Spec}(k)$ is the blow-up of $X$ along $Z$, we have, in $\ell$-adic cohomology:

$$(*)\ \ H^a(X',\mathbf{Z}_{\ell}(n)) = H^a(X,\mathbf{Z}_{\ell}(n))\oplus\bigoplus_{j=1}^{c-1}H^{a-2j}(Z,\mathbf{Z}_{\ell}(n-j)).$$

Is there, in addition, a splitting of the complex $Rf'_*\mathbf{Z}_{\ell}(n)$ in the derived category of $\ell$-adic sheaves? What is this splitting?

It should be something like

$$Rf'_*\mathbf{Z}_{\ell}(n) = \bigoplus_j \left(H^j(X,\mathbf{Z}_{\ell}(n))[-2j]\oplus i_*H^j(Z,\mathbf{Z}_{\ell}(n-c))[-2j+2c]\right)\ ?$$

I'd like to ask for a reference for the proof of the derived version of formula $(*)$, or an argument, with correct weights and shifts.

Thanks a lot!