The Stacks project

Lemma 36.3.9. Let $X$ be a scheme.

  1. For objects $K, L$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the derived tensor product $K \otimes ^\mathbf {L}_{\mathcal{O}_ X} L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

  2. If $X = \mathop{\mathrm{Spec}}(A)$ is affine then

    \[ \widetilde{M^\bullet } \otimes _{\mathcal{O}_ X}^\mathbf {L} \widetilde{K^\bullet } = \widetilde{M^\bullet \otimes _ A^\mathbf {L} K^\bullet } \]

    for any pair of complexes of $A$-modules $K^\bullet $, $M^\bullet $.

Proof. The equality of (2) follows immediately from Lemma 36.3.6 and the construction of the derived tensor product. To see (1) let $K, L$ be objects of $D_\mathit{QCoh}(\mathcal{O}_ X)$. To check that $K \otimes ^\mathbf {L} L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we may work locally on $X$, hence we may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine. By Lemma 36.3.5 we may represent $K$ and $L$ by complexes of $A$-modules. Then part (2) implies the result. $\square$

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Your email address will not be published.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 08DX. Beware of the difference between the letter 'O' and the digit '0'.