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].