Lemma 13.38.1.reference Let \mathcal{D} be a triangulated category with direct sums which is compactly generated. Let H : \mathcal{D} \to \textit{Ab} be a contravariant cohomological functor which transforms direct sums into products. Then H is representable.
13.38 Brown representability
A reference for the material in this section is [Neeman-Grothendieck].
Proof. Let E_ i, i \in I be a set of compact objects such that \bigoplus _{i \in I} E_ i generates \mathcal{D}. We may and do assume that the set of objects \{ E_ i\} is preserved under shifts. Consider pairs (i, a) where i \in I and a \in H(E_ i) and set
X_1 = \bigoplus \nolimits _{(i, a)} E_ i
Since H(X_1) = \prod _{(i, a)} H(E_ i) we see that (a)_{(i, a)} defines an element a_1 \in H(X_1). Set H_1 = \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(- , X_1). By Yoneda's lemma (Categories, Lemma 4.3.5) the element a_1 defines a natural transformation H_1 \to H.
We are going to inductively construct X_ n and transformations a_ n : H_ n \to H where H_ n = \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, X_ n). Namely, we apply the procedure above to the functor \mathop{\mathrm{Ker}}(H_ n \to H) to get an object
K_{n + 1} = \bigoplus \nolimits _{(i, k),\ k \in \mathop{\mathrm{Ker}}(H_ n(E_ i) \to H(E_ i))} E_ i
and a transformation \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, K_{n + 1}) \to \mathop{\mathrm{Ker}}(H_ n \to H). By Yoneda's lemma the composition \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, K_{n + 1}) \to H_ n gives a morphism K_{n + 1} \to X_ n. We choose a distinguished triangle
K_{n + 1} \to X_ n \to X_{n + 1} \to K_{n + 1}[1]
in \mathcal{D}. The element a_ n \in H(X_ n) maps to zero in H(K_{n + 1}) by construction. Since H is cohomological we can lift it to an element a_{n + 1} \in H(X_{n + 1}).
We claim that X = \text{hocolim} X_ n represents H. Applying H to the defining distinguished triangle
\bigoplus X_ n \to \bigoplus X_ n \to X \to \bigoplus X_ n[1]
we obtain an exact sequence
\prod H(X_ n) \leftarrow \prod H(X_ n) \leftarrow H(X)
Thus there exists an element a \in H(X) mapping to (a_ n) in \prod H(X_ n). Hence a natural transformation \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(- , X) \to H such that
\mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, X_1) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, X_2) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, X_3) \to \ldots \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, X) \to H
commutes. For each i the map \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(E_ i, X) \to H(E_ i) is surjective, by construction of X_1. On the other hand, by construction of X_ n \to X_{n + 1} the kernel of \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(E_ i, X_ n) \to H(E_ i) is killed by the map \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(E_ i, X_ n) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(E_ i, X_{n + 1}). Since
\mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(E_ i, X) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(E_ i, X_ n)
by Lemma 13.33.9 we see that \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(E_ i, X) \to H(E_ i) is injective.
To finish the proof, consider the subcategory
\mathcal{D}' = \{ Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}) \mid \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(Y[n], X) \to H(Y[n]) \text{ is an isomorphism for all }n\}
As \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, X) \to H is a transformation between cohomological functors, the subcategory \mathcal{D}' is a strictly full, saturated, triangulated subcategory of \mathcal{D} (details omitted; see proof of Lemma 13.6.3). Moreover, as both H and \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(-, X) transform direct sums into products, we see that direct sums of objects of \mathcal{D}' are in \mathcal{D}'. Thus derived colimits of objects of \mathcal{D}' are in \mathcal{D}'. Since \{ E_ i\} is preserved under shifts, we see that E_ i is an object of \mathcal{D}' for all i. It follows from Lemma 13.37.3 that \mathcal{D}' = \mathcal{D} and the proof is complete. \square
Proposition 13.38.2.reference Let \mathcal{D} be a triangulated category with direct sums which is compactly generated. Let F : \mathcal{D} \to \mathcal{D}' be an exact functor of triangulated categories which transforms direct sums into direct sums. Then F has an exact right adjoint.
Proof. For an object Y of \mathcal{D}' consider the contravariant functor
\mathcal{D} \to \textit{Ab},\quad W \mapsto \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(F(W), Y)
This is a cohomological functor as F is exact and transforms direct sums into products as F transforms direct sums into direct sums. Thus by Lemma 13.38.1 we find an object X of \mathcal{D} such that \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(W, X) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(F(W), Y). The existence of the adjoint follows from Categories, Lemma 4.24.2. Exactness follows from Lemma 13.7.1. \square
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