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 $\prod I_ n^\bullet [-n]$ is quasi-isomorphic to $E$. This works more generally in $D(\mathcal{A})$ where $\mathcal{A}$ is a Grothendieck abelian category with Ab4*.

Comments (2)

Comment #7835 by DatPham on

In the second-to-last sentence, I think it should be .

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