Lemma 46.7.6 (06ZW)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 46: Adequate Modules
Section 46.7: Derived categories of adequate modules, I
Lemma 46.7.6 (cite)

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Lemma 46.7.6. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $\mathcal{F}$ and $\mathcal{G}$ be adequate $\mathcal{O}$ -modules. For any $i \geq 0$ the natural map

$\mathop{\mathrm{Ext}}\nolimits ^ i_{\textit{Adeq}(\mathcal{O})}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\textit{Mod}(\mathcal{O})}(\mathcal{F}, \mathcal{G})$

is an isomorphism.

Proof. By definition these ext groups are computed as hom sets in the derived category. Hence this follows immediately from Lemma 46.7.5. $\square$

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