Lemma 46.4.2 (06Z7)—The Stacks project

Lemma 46.4.2. Let $A$ be a ring. Denote $\mathcal{P}$ the category of module-valued functors on $\textit{Alg}_ A$ and $\mathcal{A}$ the category of adequate functors on $\textit{Alg}_ A$. Denote $i : \mathcal{A} \to \mathcal{P}$ the inclusion functor. Denote $Q : \mathcal{P} \to \mathcal{A}$ the construction of Lemma 46.4.1. Then

$i$ is fully faithful, exact, and its image is a weak Serre subcategory,
$\mathcal{P}$ has enough injectives,
the functor $Q$ is a right adjoint to $i$ hence left exact,
$Q$ transforms injectives into injectives,
$\mathcal{A}$ has enough injectives.

Proof. This lemma just collects some facts we have already seen so far. Part (1) is clear from the definitions, the characterization of weak Serre subcategories (see Homology, Lemma 12.10.3), and Lemmas 46.3.10, 46.3.11, and 46.3.16. Recall that $\mathcal{P}$ is equivalent to the category $\textit{PMod}((\textit{Aff}/\mathop{\mathrm{Spec}}(A))_\tau , \mathcal{O})$. Hence (2) by Injectives, Proposition 19.8.5. Part (3) follows from Lemma 46.4.1 and Categories, Lemma 4.24.5. Parts (4) and (5) follow from Homology, Lemmas 12.29.1 and 12.29.3. $\square$

The Stacks project

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