Lemma 48.33.1 (0G5A)—The Stacks project

Lemma 48.33.1. Let $p : U \to \mathop{\mathrm{Spec}}(k)$ be separated of finite type where $k$ is a field. Let $\omega _{U/k}^\bullet = p^!\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}$. There are canonical isomorphisms

\[ \mathop{\mathrm{Hom}}\nolimits _ k(H^ i(U, K), k) = H^{-i}_ c(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(K, \omega _{U/k}^\bullet )) \]

of topological $k$-vector spaces functorial for $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$.

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

\[ H^ i(U, K) = \mathop{\mathrm{Ext}}\nolimits ^ i_ U(\mathcal{O}_ U, M|_ U) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{I}^ n, M) = \mathop{\mathrm{colim}}\nolimits H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, M)) \]

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

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ k( & H^ i(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, M)), k) \\ & = \mathop{\mathrm{Ext}}\nolimits ^{-i}_ X(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, M), \omega _{X/k}^\bullet ) \\ & = H^{-i}(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}( \mathcal{I}^ n, M), \omega _{X/k}^\bullet )) \end{align*}

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

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(M, \omega _{X/k}^\bullet ) \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{I}^ n \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}( \mathcal{I}^ n, M), \omega _{X/k}^\bullet ) \]

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

\[ \mathop{\mathrm{lim}}\nolimits H^{-i}(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(M, \omega _{X/k}^\bullet ) \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{I}^ n) \]

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. $\square$

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