Lemma 48.2.3 (0A88)—The Stacks project

Lemma 48.2.3. Let $A$ be a Noetherian ring and let $X = \mathop{\mathrm{Spec}}(A)$. Let $K, L$ be objects of $D(A)$. If $K \in D_{\textit{Coh}}(A)$ and $L$ has finite injective dimension, then

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\widetilde{K}, \widetilde{L}) = \widetilde{R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)} \]

in $D(\mathcal{O}_ X)$.

Proof. 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. In this case, Derived Categories of Schemes, Lemma 36.10.8, tells us that the $n$th cohomology sheaf of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\widetilde{K}, \widetilde{L})$ is the sheaf associated to the presheaf

\[ D(f) \longmapsto \mathop{\mathrm{Ext}}\nolimits ^ n_{A_ f}(K \otimes _ A A_ f, I \otimes _ A A_ f) \]

Since $A$ is Noetherian, the $A_ f$-module $I \otimes _ A A_ f$ is injective (Dualizing Complexes, Lemma 47.3.8). Hence we see that

\begin{align*} \mathop{\mathrm{Ext}}\nolimits ^ n_{A_ f}(K \otimes _ A A_ f, I \otimes _ A A_ f) & = \mathop{\mathrm{Hom}}\nolimits _{A_ f}(H^{-n}(K \otimes _ A A_ f), I \otimes _ A A_ f) \\ & = \mathop{\mathrm{Hom}}\nolimits _{A_ f}(H^{-n}(K) \otimes _ A A_ f, I \otimes _ A A_ f) \\ & = \mathop{\mathrm{Hom}}\nolimits _ A(H^{-n}(K), I) \otimes _ A A_ f \end{align*}

The last equality because $H^{-n}(K)$ is a finite $A$-module, see Algebra, Lemma 10.10.2. This proves that the canonical map

\[ \widetilde{R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)} \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\widetilde{K}, \widetilde{L}) \]

is a quasi-isomorphism in this case and the proof is done. $\square$

The Stacks project

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