The Stacks project

Lemma 15.98.2. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. the map

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

of Lemma 15.73.3 is an isomorphism if the following three conditions are satisfied

  1. $L, M$ have finite injective dimension,

  2. $R\mathop{\mathrm{Hom}}\nolimits _ R(L, M)$ has finite tor dimension,

  3. for every $n \in \mathbf{Z}$ the truncation $\tau _{\leq n}K$ is pseudo-coherent

Proof. Pick an integer $n$ and consider the distinguished triangle

\[ \tau _{\leq n}K \to K \to \tau _{\geq n + 1}K \to \tau _{\leq n}K[1] \]

see Derived Categories, Remark 13.12.4. By assumption (3) and Lemma 15.98.1 the map is an isomorphism for $\tau _{\leq n}K$. Hence it suffices to show that both

\[ R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} \tau _{\geq n + 1}K \quad \text{and}\quad R\mathop{\mathrm{Hom}}\nolimits _ R(R\mathop{\mathrm{Hom}}\nolimits _ R(\tau _{\geq n + 1}K, L), M) \]

have vanishing cohomology in degrees $\leq n - c$ for some $c$. This follows immediately from assumptions (2) and (1). $\square$

Comments (0)

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Your email address will not be published.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).

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A69. Beware of the difference between the letter 'O' and the digit '0'.