Lemma 19.14.4 (0F5V)—The Stacks project

Lemma 19.14.4. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $R$ , $F$ , $G$ be as in the Gabriel-Popescu theorem (Theorem 19.14.3). Then we obtain derived functors

$RG : D(\mathcal{A}) \to D(\text{Mod}_ R) \quad \text{and}\quad F : D(\text{Mod}_ R) \to D(\mathcal{A})$

such that $F$ is left adjoint to $RG$ , $RG$ is fully faithful, and $F \circ RG = \text{id}$ .

Proof. The existence and adjointness of the functors follows from Theorems 19.14.3 and 19.12.6 and Derived Categories, Lemmas 13.31.6, 13.16.9, and 13.30.3. The statement $F \circ RG = \text{id}$ follows because we can compute $RG$ on an object of $D(\mathcal{A})$ by applying $G$ to a suitable representative complex $I^\bullet$ (for example a K-injective one) and then $F(G(I^\bullet )) = I^\bullet$ because $F \circ G = \text{id}$ . Fully faithfulness of $RG$ follows from this by Categories, Lemma 4.24.4. $\square$

The Stacks project

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