The Stacks project

Lemma 13.30.1. In the situation above assume $F$ is right adjoint to $G$. Let $K \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ and $M \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}')$. If $RF$ is defined at $K$ and $LG$ is defined at $M$, then there is a canonical isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _{(S')^{-1}\mathcal{D}'}(M, RF(K)) = \mathop{\mathrm{Hom}}\nolimits _{S^{-1}\mathcal{D}}(LG(M), K) \]

This isomorphism is functorial in both variables on the triangulated subcategories of $S^{-1}\mathcal{D}$ and $(S')^{-1}\mathcal{D}'$ where $RF$ and $LG$ are defined.

Proof. Since $RF$ is defined at $K$, we see that the rule which assigns to an $s : K \to I$ in $S$ the object $F(I)$ is essentially constant as an ind-object of $(S')^{-1}\mathcal{D}'$ with value $RF(K)$. Similarly, the rule which assigns to a $t : P \to M$ in $S'$ the object $G(P)$ is essentially constant as a pro-object of $S^{-1}\mathcal{D}$ with value $LG(M)$. Thus we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{(S')^{-1}\mathcal{D}'}(M, RF(K)) & = \mathop{\mathrm{colim}}\nolimits _{s : K \to I} \mathop{\mathrm{Hom}}\nolimits _{(S')^{-1}\mathcal{D}'}(M, F(I)) \\ & = \mathop{\mathrm{colim}}\nolimits _{s : K \to I} \mathop{\mathrm{colim}}\nolimits _{t : P \to M} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(P, F(I)) \\ & = \mathop{\mathrm{colim}}\nolimits _{t : P \to M} \mathop{\mathrm{colim}}\nolimits _{s : K \to I} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(P, F(I)) \\ & = \mathop{\mathrm{colim}}\nolimits _{t : P \to M} \mathop{\mathrm{colim}}\nolimits _{s : K \to I} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}}(G(P), I) \\ & = \mathop{\mathrm{colim}}\nolimits _{t : P \to M} \mathop{\mathrm{Hom}}\nolimits _{S^{-1}\mathcal{D}}(G(P), K) \\ & = \mathop{\mathrm{Hom}}\nolimits _{S^{-1}\mathcal{D}}(LG(M), K) \end{align*}

The first equality holds by Categories, Lemma 4.22.9. The second equality holds by the definition of morphisms in $D(\mathcal{B})$, see Categories, Remark 4.27.15. The third equality holds by Categories, Lemma 4.14.10. The fourth equality holds because $F$ and $G$ are adjoint. The fifth equality holds by definition of morphism in $D(\mathcal{A})$, see Categories, Remark 4.27.7. The sixth equality holds by Categories, Lemma 4.22.10. We omit the proof of functoriality. $\square$

Comments (2)

Comment #7791 by Heiko Braun on

typo: "an s" in the first line of the proof

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