Definition 21.41.1. Let $\mathcal{C}$ be a site. Let $\mathcal{A}_\bullet $ be a simplicial sheaf of rings on $\mathcal{C}$. A simplicial $\mathcal{A}_\bullet $-module $\mathcal{F}_\bullet $ (sometimes called a simplicial sheaf of $\mathcal{A}_\bullet $-modules) is a sheaf of modules over the sheaf of rings on $\Delta \times \mathcal{C}$ associated to $\mathcal{A}_\bullet $.
21.41 Simplicial modules
Let $A_\bullet $ be a simplicial ring. Recall that we may think of $A_\bullet $ as a sheaf on $\Delta $ (endowed with the chaotic topology), see Simplicial, Section 14.4. Then a simplicial module $M_\bullet $ over $A_\bullet $ is just a sheaf of $A_\bullet $-modules on $\Delta $. In other words, for every $n \geq 0$ we have an $A_ n$-module $M_ n$ and for every map $\varphi : [n] \to [m]$ we have a corresponding map
\[ M_\bullet (\varphi ) : M_ m \longrightarrow M_ n \]
which is $A_\bullet (\varphi )$-linear such that these maps compose in the usual manner.
Let $\mathcal{C}$ be a site. A simplicial sheaf of rings $\mathcal{A}_\bullet $ on $\mathcal{C}$ is a simplicial object in the category of sheaves of rings on $\mathcal{C}$. In this case the assignment $U \mapsto \mathcal{A}_\bullet (U)$ is a sheaf of simplicial rings and in fact the two notions are equivalent. A similar discussion holds for simplicial abelian sheaves, simplicial sheaves of Lie algebras, and so on.
However, as in the case of simplicial rings above, there is another way to think about simplicial sheaves. Namely, consider the projection
\[ p : \Delta \times \mathcal{C} \longrightarrow \mathcal{C} \]
This defines a fibred category with strongly cartesian morphisms exactly the morphisms of the form $([n], U) \to ([n], V)$. We endow the category $\Delta \times \mathcal{C}$ with the topology inherited from $\mathcal{C}$ (see Stacks, Section 8.10). The simple description of the coverings in $\Delta \times \mathcal{C}$ (Stacks, Lemma 8.10.1) immediately implies that a simplicial sheaf of rings on $\mathcal{C}$ is the same thing as a sheaf of rings on $\Delta \times \mathcal{C}$.
By analogy with the case of simplicial modules over a simplicial ring, we define simplicial modules over simplicial sheaves of rings as follows.
We obtain a category $\textit{Mod}(\mathcal{A}_\bullet )$ of simplicial modules and a corresponding derived category $D(\mathcal{A}_\bullet )$. Given a map $\mathcal{A}_\bullet \to \mathcal{B}_\bullet $ of simplicial sheaves of rings we obtain a functor
\[ - \otimes ^\mathbf {L}_{\mathcal{A}_\bullet } \mathcal{B}_\bullet : D(\mathcal{A}_\bullet ) \longrightarrow D(\mathcal{B}_\bullet ) \]
Moreover, the material of the preceding sections determines a functor
\[ L\pi _! : D(\mathcal{A}_\bullet ) \longrightarrow D(\mathcal{C}) \]
Given a simplicial module $\mathcal{F}_\bullet $ the object $L\pi _!(\mathcal{F}_\bullet )$ is represented by the associated chain complex $s(\mathcal{F}_\bullet )$ (Simplicial, Section 14.23). This follows from Lemmas 21.40.2 and 21.39.7.
Lemma 21.41.2. Let $\mathcal{C}$ be a site. Let $\mathcal{A}_\bullet \to \mathcal{B}_\bullet $ be a homomorphism of simplicial sheaves of rings on $\mathcal{C}$. If $L\pi _!\mathcal{A}_\bullet \to L\pi _!\mathcal{B}_\bullet $ is an isomorphism in $D(\mathcal{C})$, then we have for all $K$ in $D(\mathcal{A}_\bullet )$.
\[ L\pi _!(K) = L\pi _!(K \otimes ^\mathbf {L}_{\mathcal{A}_\bullet } \mathcal{B}_\bullet ) \]
Proof. Let $([n], U)$ be an object of $\Delta \times \mathcal{C}$. Since $L\pi _!$ commutes with colimits, it suffices to prove this for bounded above complexes of $\mathcal{O}$-modules (compare with argument of Derived Categories, Proposition 13.29.2 or just stick to bounded above complexes). Every such complex is quasi-isomorphic to a bounded above complex whose terms are flat modules, see Modules on Sites, Lemma 18.28.8. Thus it suffices to prove the lemma for a flat $\mathcal{A}_\bullet $-module $\mathcal{F}$. In this case the derived tensor product is the usual tensor product and is a sheaf also. Hence by Lemma 21.40.2 we can compute the cohomology sheaves of both sides of the equation by the procedure of Lemma 21.40.1. Thus it suffices to prove the result for the restriction of $\mathcal{F}$ to the fibre categories (i.e., to $\Delta \times U$). In this case the result follows from Lemma 21.39.12. $\square$
Remark 21.41.3. Let $\mathcal{C}$ be a site. Let $\epsilon : \mathcal{A}_\bullet \to \mathcal{O}$ be an augmentation (Simplicial, Definition 14.20.1) in the category of sheaves of rings. Assume $\epsilon $ induces a quasi-isomorphism $s(\mathcal{A}_\bullet ) \to \mathcal{O}$. In this case we obtain an exact functor of triangulated categories Namely, for any object $K$ of $D(\mathcal{A}_\bullet )$ we have $L\pi _!(K) = L\pi _!(K \otimes _{\mathcal{A}_\bullet }^\mathbf {L} \mathcal{O})$ by Lemma 21.41.2. Thus we can define the displayed functor as the composition of $- \otimes ^\mathbf {L}_{\mathcal{A}_\bullet } \mathcal{O}$ with the functor $L\pi _! : D(\Delta \times \mathcal{C}, \pi ^{-1}\mathcal{O}) \to D(\mathcal{O})$ of Remark 21.38.6. In other words, we obtain a $\mathcal{O}$-module structure on $L\pi _!(K)$ coming from the (canonical, functorial) identification of $L\pi _!(K)$ with $L\pi _!(K \otimes _{\mathcal{A}_\bullet }^\mathbf {L} \mathcal{O})$ of the lemma.
\[ L\pi _! : D(\mathcal{A}_\bullet ) \longrightarrow D(\mathcal{O}) \]
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