The Stacks project

46.9 Higher exts of quasi-coherent sheaves on the big site

It turns out that the module-valued functor $\underline{I}$ associated to a pure injective module $I$ gives rise to an injective object in the category of adequate functors on $\textit{Alg}_ A$. Warning: It is not true that a pure projective module gives rise to a projective object in the category of adequate functors. We do have plenty of projective objects, namely, the linearly adequate functors.

Lemma 46.9.1. Let $A$ be a ring. Let $\mathcal{A}$ be the category of adequate functors on $\textit{Alg}_ A$. The injective objects of $\mathcal{A}$ are exactly the functors $\underline{I}$ where $I$ is a pure injective $A$-module.

Proof. Let $I$ be an injective object of $\mathcal{A}$. Choose an embedding $I \to \underline{M}$ for some $A$-module $M$. As $I$ is injective we see that $\underline{M} = I \oplus F$ for some module-valued functor $F$. Then $M = I(A) \oplus F(A)$ and it follows that $I = \underline{I(A)}$. Thus we see that any injective object is of the form $\underline{I}$ for some $A$-module $I$. It is clear that the module $I$ has to be pure injective since any universally exact sequence $0 \to M \to N \to L \to 0$ gives rise to an exact sequence $0 \to \underline{M} \to \underline{N} \to \underline{L} \to 0$ of $\mathcal{A}$.

Finally, suppose that $I$ is a pure injective $A$-module. Choose an embedding $\underline{I} \to J$ into an injective object of $\mathcal{A}$ (see Lemma 46.4.2). We have seen above that $J = \underline{I'}$ for some $A$-module $I'$ which is pure injective. As $\underline{I} \to \underline{I'}$ is injective the map $I \to I'$ is universally injective. By assumption on $I$ it splits. Hence $\underline{I}$ is a summand of $J = \underline{I'}$ whence an injective object of the category $\mathcal{A}$. $\square$

Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $M$ be an $A$-module. We will use the notation $M^ a$ to denote the quasi-coherent sheaf of $\mathcal{O}$-modules on $(\mathit{Sch}/U)_\tau $ associated to the quasi-coherent sheaf $\widetilde{M}$ on $U$. Now we have all the notation in place to formulate the following lemma.

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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