Lemma 15.73.5. Let $R$ be a ring. Given complexes $K, L, M$ in $D(R)$ there is a canonical morphism

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

Lemma 15.73.5. Let $R$ be a ring. Given complexes $K, L, M$ in $D(R)$ there is a canonical morphism

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

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

**Proof.**
Choose a K-flat complex $K^\bullet $ representing $K$, and a K-injective complex $I^\bullet $ representing $L$, and choose any complex $M^\bullet $ representing $M$. Choose a quasi-isomorphism $\text{Tot}(K^\bullet \otimes _ R I^\bullet ) \to J^\bullet $ where $J^\bullet $ is K-injective. Then we use the map

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

where the first map is the map from Lemma 15.71.4. $\square$

Your email address will not be published.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)

There are also: