The Stacks project

Lemma 46.9.2. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $M$, $N$ be $A$-modules. For all $i$ we have a canonical isomorphism

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\textit{Mod}(\mathcal{O})}(M^ a, N^ a) = \text{Pext}^ i_ A(M, N) \]

functorial in $M$ and $N$.

Proof. Let us construct a canonical arrow from right to left. Namely, if $N \to I^\bullet $ is a pure injective resolution, then $M^ a \to (I^\bullet )^ a$ is an exact complex of (adequate) $\mathcal{O}$-modules. Hence any element of $\text{Pext}^ i_ A(M, N)$ gives rise to a map $N^ a \to M^ a[i]$ in $D(\mathcal{O})$, i.e., an element of the group on the left.

To prove this map is an isomorphism, note that we may replace $\mathop{\mathrm{Ext}}\nolimits ^ i_{\textit{Mod}(\mathcal{O})}(M^ a, N^ a)$ by $\mathop{\mathrm{Ext}}\nolimits ^ i_{\textit{Adeq}(\mathcal{O})}(M^ a, N^ a)$, see Lemma 46.7.6. Let $\mathcal{A}$ be the category of adequate functors on $\textit{Alg}_ A$. We have seen that $\mathcal{A}$ is equivalent to $\textit{Adeq}(\mathcal{O})$, see Lemma 46.5.3; see also the proof of Lemma 46.7.3. Hence now it suffices to prove that

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {A}(\underline{M}, \underline{N}) = \text{Pext}^ i_ A(M, N) \]

However, this is clear from Lemma 46.9.1 as a pure injective resolution $N \to I^\bullet $ exactly corresponds to an injective resolution of $\underline{N}$ in $\mathcal{A}$. $\square$

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