The Stacks project

Remark 21.39.11 (Simplicial modules). Let $\mathcal{C} = \Delta $ and let $B$ be any ring. This is a special case of Example 21.39.1 where the assumptions of Lemma 21.39.7 hold. Namely, let $U_\bullet $ be the cosimplicial object of $\Delta $ given by the identity functor. To verify the condition we have to show that for $[m] \in \mathop{\mathrm{Ob}}\nolimits (\Delta )$ the simplicial set $\Delta [m] : n \mapsto \mathop{\mathrm{Mor}}\nolimits _\Delta ([n], [m])$ is homotopy equivalent to a point. This is explained in Simplicial, Example 14.26.7.

In this situation the category $\textit{Mod}(\underline{B})$ is just the category of simplicial $B$-modules and the functor $L\pi _!$ sends a simplicial $B$-module $M_\bullet $ to its associated complex $s(M_\bullet )$ of $B$-modules. Thus the results above can be reinterpreted in terms of results on simplicial modules. For example a special case of Lemma 21.39.10 is: if $M_\bullet $, $M'_\bullet $ are flat simplicial $B$-modules, then the complex $s(M_\bullet \otimes _ B M'_\bullet )$ is quasi-isomorphic to the total complex associated to the double complex $s(M_\bullet ) \otimes _ B s(M'_\bullet )$. (Hint: use flatness to convert from derived tensor products to usual tensor products.) This is a special case of the Eilenberg-Zilber theorem which can be found in [Eilenberg-Zilber].


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Your email address will not be published.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