Lemma 86.2.3. Let $A$ be a Noetherian ring and let $X = \mathop{\mathrm{Spec}}(A)$. Let $\mathcal{O}_{\acute{e}tale}$ be the structure sheaf of $X$ on the small étale site of $X$. Let $K, L$ be objects of $D(A)$. If $K \in D_{\textit{Coh}}(A)$ and $L$ has finite injective dimension, then
\[ \epsilon ^*\widetilde{R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)} = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\acute{e}tale}}(\epsilon ^*\widetilde{K}, \epsilon ^*\widetilde{L}) \]
in $D(\mathcal{O}_{\acute{e}tale})$ where $\epsilon : (X_{\acute{e}tale}, \mathcal{O}_{\acute{e}tale}) \to (X, \mathcal{O}_ X)$ is as in Derived Categories of Spaces, Section 75.4.
Proof.
By Duality for Schemes, Lemma 48.2.3 we have a canonical isomorphism
\[ \widetilde{R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)} = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\widetilde{K}, \widetilde{L}) \]
in $D(\mathcal{O}_ X)$. There is a canonical map
\[ \epsilon ^*R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\widetilde{K}, \widetilde{L}) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\acute{e}tale}}(\epsilon ^*\widetilde{K}, \epsilon ^*\widetilde{L}) \]
in $D(\mathcal{O}_{\acute{e}tale})$, see Cohomology on Sites, Remark 21.35.11. We will show the left and right hand side of this arrow have isomorphic cohomology sheaves, but we will omit the verification that the isomorphism is given by this arrow.
We may assume that $L$ is given by a finite complex $I^\bullet $ of injective $A$-modules. By induction on the length of $I^\bullet $ and compatibility of the constructions with distinguished triangles, we reduce to the case that $L = I[0]$ where $I$ is an injective $A$-module. Recall that the cohomology sheaves of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\acute{e}tale}}(\epsilon ^*\widetilde{K}, \epsilon ^*\widetilde{L}))$ are the sheafifications of the presheaf sending $U$ étale over $X$ to the $i$th ext group between the restrictions of $\epsilon ^*\widetilde{K}$ and $\epsilon ^*\widetilde{L}$ to $U_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.35.1. If $U = \mathop{\mathrm{Spec}}(B)$ is affine, then this ext group is equal to $\text{Ext}^ i_ B(K \otimes _ A B, L \otimes _ A B)$ by the equivalence of Derived Categories of Spaces, Lemma 75.4.2 and Derived Categories of Schemes, Lemma 36.3.5 (this also uses the compatibilities detailed in Derived Categories of Spaces, Remark 75.6.3). Since $A \to B$ is étale, we see that $I \otimes _ A B$ is an injective $B$-module by Dualizing Complexes, Lemma 47.26.4. Hence we see that
\begin{align*} \mathop{\mathrm{Ext}}\nolimits ^ n_ B(K \otimes _ A B, I \otimes _ A B) & = \mathop{\mathrm{Hom}}\nolimits _ B(H^{-n}(K \otimes _ A B), I \otimes _ A B) \\ & = \mathop{\mathrm{Hom}}\nolimits _{A_ f}(H^{-n}(K) \otimes _ A B, I \otimes _ A B) \\ & = \mathop{\mathrm{Hom}}\nolimits _ A(H^{-n}(K), I) \otimes _ A B \\ & = \text{Ext}^ n_ A(K, I) \otimes _ A B \end{align*}
The penultimate equality because $H^{-n}(K)$ is a finite $A$-module, see More on Algebra, Lemma 15.65.4. Therefore the cohomology sheaves of the left and right hand side of the equality in the lemma are the same.
$\square$
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