The Stacks project

Remark 19.13.5. Let $R$ be a ring. Suppose that $M_ n$, $n \in \mathbf{Z}$ are $R$-modules. Denote $E_ n = M_ n[-n] \in D(R)$. We claim that $E = \bigoplus M_ n[-n]$ is both the direct sum and the product of the objects $E_ n$ in $D(R)$. To see that it is the direct sum, take a look at the proof of Lemma 19.13.4. To see that it is the direct product, take injective resolutions $M_ n \to I_ n^\bullet $. By the proof of Lemma 19.13.4 we have

\[ \prod E_ n = \prod I_ n^\bullet [-n] \]

in $D(R)$. Since products in $\text{Mod}_ R$ are exact, we see that $\prod I_ n^\bullet $ is quasi-isomorphic to $E$. This works more generally in $D(\mathcal{A})$ where $\mathcal{A}$ is a Grothendieck abelian category with Ab4*.


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Your email address will not be published.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 07DA. Beware of the difference between the letter 'O' and the digit '0'.