Proposition 19.15.2. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $\mathcal{D}$ be a triangulated category. Let $F : D(\mathcal{A}) \to \mathcal{D}$ 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
\[ D(\mathcal{A}) \to \textit{Ab},\quad W \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(F(W), Y) \]
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 19.15.1 we find an object $X$ of $D(\mathcal{A})$ such that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(W, X) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(F(W), Y)$. The existence of the adjoint follows from Categories, Lemma 4.24.2. Exactness follows from Derived Categories, Lemma 13.7.1. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)