Lemma 63.16.1. Let $f : X \to Y$ be a separated finite type morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a torsion ring. Let $K \in D(Y_{\acute{e}tale}, \Lambda )$. For $n \in \mathbf{Z}$ the cohomology sheaf $H^ n(Rf^!K)$ restricted to $X_{affine, {\acute{e}tale}}$ is the sheaf associated to the presheaf

\[ U \longmapsto \mathop{\mathrm{Hom}}\nolimits _ Y(R(U \to Y)_!\Lambda , K[n]) \]

See discussion above for the functorial nature of $R(U \to Y)_!\Lambda $.

**Proof.**
Let $j : U \to X$ be an object of $X_{affine, {\acute{e}tale}}$ and set $g = f \circ j$. Recall that $\mathop{\mathrm{Hom}}\nolimits _ X(j_!\Lambda , M[n]) = H^ n(U, M)$ for any $M$ in $D(X_{\acute{e}tale}, \Lambda )$. Then $H^ n(Rf^!K)$ is the sheaf associated to the presheaf

\[ U \mapsto H^ n(U, Rf^!K) = \mathop{\mathrm{Hom}}\nolimits _ X(j_!\Lambda , Rf^!K[n]) = \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!j_!\Lambda , K[n] = \mathop{\mathrm{Hom}}\nolimits _ Y(Rg_!\Lambda , K[n]) \]

We omit the verification that the transition maps are given by the transition maps between the objects $Rg_!\Lambda = R(U \to Y)_!\Lambda $ we constructed above.
$\square$

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