Lemma 15.73.3. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical morphism

in $D(R)$ functorial in $K, L, M$.

Lemma 15.73.3. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical morphism

\[ R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(R\mathop{\mathrm{Hom}}\nolimits _ R(K, L), M) \]

in $D(R)$ functorial in $K, L, M$.

**Proof.**
Choose a K-injective complex $I^\bullet $ representing $M$, a K-injective complex $J^\bullet $ representing $L$, and a K-flat complex $K^\bullet $ representing $K$. The map is defined using the map

\[ \text{Tot}(\mathop{\mathrm{Hom}}\nolimits ^\bullet (J^\bullet , I^\bullet ) \otimes _ R K^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , J^\bullet ), I^\bullet ) \]

of Lemma 15.71.6. We omit the proof that this is functorial in all three objects of $D(R)$. $\square$

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)