Lemma 15.68.1. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical isomorphism

in $D(R)$ functorial in $K, L, M$ which recovers (15.68.0.1) by taking $H^0$.

Lemma 15.68.1. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical isomorphism

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

in $D(R)$ functorial in $K, L, M$ which recovers (15.68.0.1) by taking $H^0$.

**Proof.**
Choose a K-injective complex $I^\bullet $ representing $M$ and a K-flat complex of $R$-modules $L^\bullet $ representing $L$. For any complex of $R$-modules $K^\bullet $ we have

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

by Lemma 15.67.1. The lemma follows by the definition of $R\mathop{\mathrm{Hom}}\nolimits $ and because $\text{Tot}(K^\bullet \otimes _ R L^\bullet )$ represents the derived tensor product. $\square$

Your email address will not be published.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)