Lemma 19.15.1. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $H : D(\mathcal{A}) \to \textit{Ab}$ be a contravariant cohomological functor which transforms direct sums into products. Then $H$ is representable.
Proof. Let $R, F, G, RG$ be as in Lemma 19.14.4 and consider the functor $H \circ F : D(\text{Mod}_ R) \to \textit{Ab}$. Observe that since $F$ is a left adjoint it sends direct sums to direct sums and hence $H \circ F$ transforms direct sums into products. On the other hand, the derived category $D(\text{Mod}_ R)$ is generated by a single compact object, namely $R$. By Derived Categories, Lemma 13.38.1 we see that $H \circ F$ is representable, say by $L \in D(\text{Mod}_ R)$. Choose a distinguished triangle
\[ M \to L \to RG(F(L)) \to M[1] \]
in $D(\text{Mod}_ R)$. Then $F(M) = 0$ because $F \circ RG = \text{id}$. Hence $H(F(M)) = 0$ hence $\mathop{\mathrm{Hom}}\nolimits (M, L) = 0$. It follows that $L \to RG(F(L))$ is the inclusion of a direct summand, see Derived Categories, Lemma 13.4.11. For $A$ in $D(\mathcal{A})$ we obtain
\begin{align*} H(A) & = H(F(RG(A)) \\ & = \mathop{\mathrm{Hom}}\nolimits (RG(A), L) \\ & \to \mathop{\mathrm{Hom}}\nolimits (RG(A), RG(F(L))) \\ & = \mathop{\mathrm{Hom}}\nolimits (F(RG(A)), F(L)) \\ & = \mathop{\mathrm{Hom}}\nolimits (A, F(L)) \end{align*}
where the arrow has a left inverse functorial in $A$. In other words, we find that $H$ is the direct summand of a representable functor. Since $D(\mathcal{A})$ is Karoubian (Derived Categories, Lemma 13.4.14) we conclude. $\square$
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