The Stacks project

Lemma 15.103.2. Let

\[ (A_0^\bullet \to A_1^\bullet \to A_2^\bullet \to \ldots ) \longrightarrow (B_0^\bullet \to B_1^\bullet \to B_2^\bullet \to \ldots ) \]

be a map between two complexes of complexes of abelian groups. Set $A^{p, q} = A_ p^ q$, $B^{p, q} = B_ p^ q$ to obtain double complexes. Let $\text{Tot}_\pi (A^{\bullet , \bullet })$ and $\text{Tot}_\pi (B^{\bullet , \bullet })$ be the product total complexes associated to the double complexes. If each $A_ p^\bullet \to B_ p^\bullet $ is a quasi-isomorphism, then $\text{Tot}_\pi (A^{\bullet , \bullet }) \to \text{Tot}_\pi (B^{\bullet , \bullet })$ is a quasi-isomorphism.

Proof. Recall that $\text{Tot}_\pi (A^{\bullet , \bullet })$ in degree $n$ is given by $\prod _{p + q = n} A^{p, q} = \prod _{p + 1 = n} A^ q_ p$. Let $C_ p^\bullet $ be the cone on the map $A_ p^\bullet \to B_ p^\bullet $, see Derived Categories, Section 13.9. By the functoriality of the cone construction we obtain a complex of complexes

\[ C_0^\bullet \to C_1^\bullet \to C_2^\bullet \to \ldots \]

Then we see $\text{Tot}_\pi (C^{\bullet , \bullet })$ in degree $n$ is given by

\[ \prod _{p + q = n} C^{p, q} = \prod _{p + q = n} C^ q_ p = \prod _{p + q = n} (B^ q_ p \oplus A^{q + 1}_ p) = \prod _{p + q = n} B^ q_ p \oplus \prod _{p + q = n} A^{q + 1}_ p \]

We conclude that $\text{Tot}_\pi (C^{\bullet , \bullet })$ is the cone of the map $\text{Tot}_\pi (A^{\bullet , \bullet }) \to \text{Tot}_\pi (B^{\bullet , \bullet })$ (We omit the verification that the differentials agree.) Thus it suffices to show $\text{Tot}_\pi (A^{\bullet , \bullet })$ is acyclic if each $A_ p^\bullet $ is acyclic. This follows from Lemma 15.103.1. $\square$

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Your email address will not be published.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