The Stacks project

Proof. Choose a compactification $j : U \to X$ over $k$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent ideal sheaf with $V(\mathcal{I}) = X \setminus U$. By Derived Categories of Schemes, Proposition 36.11.2 we may choose $M \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ with $K = M|_ U$. We have

by Lemma 48.30.1. Since $\mathcal{I}^ n$ is a coherent $\mathcal{O}_ X$-module, we have $\mathcal{I}^ n$ in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$, hence $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, M)$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.11.5.

Let $\omega _{X/k}^\bullet = q^!\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}$ where $q : X \to \mathop{\mathrm{Spec}}(k)$ is the structure morphism, see Section 48.27. We find that

by Lemma 48.27.1. By Lemma 48.2.4 part (1) the canonical map

is an isomorphism. Observe that $\omega ^\bullet _{U/k} = \omega ^\bullet _{X/k}|_ U$ because $p^!$ is constructed as $q^!$ composed with restriction to $U$. Hence $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(M, \omega _{X/k}^\bullet )$ is an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ which restricts to $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(K, \omega _{U/k}^\bullet )$ on $U$. Hence by Lemma 48.30.11 we conclude that

is an avatar for the right hand side of the equality of the lemma. Combining all the isomorphisms obtained in this manner we get the isomorphism of the lemma. $\square$