Section 24.33 (0GZ8): Differential graded modules on a category

24.33 Differential graded modules on a category

This section is the continuation of Cohomology on Sites, Section 21.43.

Let $\mathcal{C}$ be a category. We think of $\mathcal{C}$ as a site with the chaotic topology. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$ . Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded $\mathcal{O}$ -algebras. In other words, $\mathcal{O}$ is a presheaf of rings on the category $\mathcal{C}$ and $(\mathcal{A}, \text{d})$ is a presheaf of differential graded $\mathcal{O}$ -algebras on $\mathcal{C}$ , see Categories, Definition 4.3.3.

Definition 24.33.1. In the situation above, we denote $\mathit{QC}(\mathcal{A}, \text{d})$ the full subcategory of $D(\mathcal{A}, \text{d})$ consisting of objects $M$ such that for all $U \to V$ in $\mathcal{C}$ the canonical map

$R\Gamma (V, M) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}(U) \longrightarrow R\Gamma (U, M)$

is an isomorphism in $D(\mathcal{A}(U), \text{d})$ .

Lemma 24.33.2. In the situation above, the subcategory $\mathit{QC}(\mathcal{A}, \text{d})$ is a strictly full, saturated, triangulated subcategory of $D(\mathcal{A}, \text{d})$ preserved by arbitrary direct sums.

Proof. Let $U$ be an object of $\mathcal{C}$ . Since the topology on $\mathcal{C}$ is chaotic, the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ is exact and commutes with direct sums. Hence the exact functor $M \mapsto R\Gamma (U, M)$ is computed by representing $K$ by any differential graded $\mathcal{A}$ -module $\mathcal{M}$ and taking $\mathcal{M}(U)$ . Thus $R\Gamma (U, -)$ commutes with direct sums, see Lemma 24.26.8. Similarly, given a morphism $U \to V$ of $\mathcal{C}$ the derived tensor product functor $- \otimes _{\mathcal{O}(A)}^\mathbf {L} \mathcal{A}(U) : D(\mathcal{A}(V)) \to D(\mathcal{A}(U))$ is exact and commutes with direct sums. The lemma follows from these observations in a straightforward manner; details omitted. $\square$

Remark 24.33.3. As above, let $\mathcal{C}$ be a category viewed as a site with the chaotic topology, let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$ , and let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded $\mathcal{O}$ -algebras. Then the analogue of Cohomology on Sites, Proposition 21.43.9 holds for $\mathit{QC}(\mathcal{A}, \text{d})$ with almost exactly the same proof:

any contravariant cohomological functor $H : \mathit{QC}(\mathcal{A}, \text{d}) \to \textit{Ab}$ which transforms direct sums into products is representable,
any exact functor $F : \mathit{QC}(\mathcal{A}, \text{d}) \to \mathcal{D}$ of triangulated categories which transforms direct sums into direct sums has an exact right adjoint, and
the inclusion functor $\mathit{QC}(\mathcal{A}, \text{d}) \to D(\mathcal{A}, \text{d})$ has an exact right adjoint.

If we ever need this we will precisely formulate and prove this here.

Let $u : \mathcal{C}' \to \mathcal{C}$ be a functor between categories. If we view $\mathcal{C}$ and $\mathcal{C}'$ as sites with the chaotic topology, then $u$ is a continuous and cocontinuous functor. Hence we obtain a morphism $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ of topoi, see Sites, Lemma 7.21.1. Additionally, suppose given sheaves of rings $\mathcal{O}$ on $\mathcal{C}$ and $\mathcal{O}'$ on $\mathcal{C}'$ and a map $g^\sharp : g^{-1}\mathcal{O} \to \mathcal{O}'$ . We denote the corresponding morphism of ringed topoi simply $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ , see Modules on Sites, Section 18.7. Finally, suppose that $(\mathcal{A}, \text{d})$ is a sheaf of differential graded $\mathcal{O}$ -algebras and that $(\mathcal{A}', \text{d})$ is a sheaf of differential graded $\mathcal{O}'$ -algebras and moreover that we are given a map $\varphi : g^*\mathcal{A} \to \mathcal{A}'$ of differential graded $\mathcal{O}'$ -algebras (see Section 24.18).

Lemma 24.33.4. Let $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ and $\varphi : g^*\mathcal{A} \to \mathcal{A}'$ be as above. Then the functor $Lg^* : D(\mathcal{A}, \text{d}) \to D(\mathcal{A}', \text{d})$ maps $\mathit{QC}(\mathcal{A}, \text{d})$ into $\mathit{QC}(\mathcal{A}', \text{d})$ .

Proof. Let $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ with image $U = u(U')$ in $\mathcal{C}$ . Let $pt$ denote the category with a single object and a single morphism. Denote $(\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}'(U'))$ and $(\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}(U))$ the ringed topoi as indicated endowed with the differential graded algebras $\mathcal{A}'(U)$ and $\mathcal{A}(U)$ . Of course we identify the derived category of differential graded modules on these with $D(\mathcal{A}'(U'), \text{d})$ and $D(\mathcal{A}(U), \text{d})$ . Then we have a commutative diagram of ringed topoi

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}'(U')) \ar[rr]_{U'} \ar[d] & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \ar[d]^ g \\ (\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}(U)) \ar[rr]^ U & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) }$

each endowed with corresponding differential graded algebras. Pullback along the lower horizontal morphism sends $M$ in $D(\mathcal{A}, \text{d})$ to $R\Gamma (U, K)$ viewed as an object in $D(\mathcal{A}(U), \text{d})$ . Pullback by the left vertical arrow sends $M$ to $M \otimes _{\mathcal{A}(U)}^\mathbf {L} \mathcal{A}'(U')$ . Going around the diagram either direction produces the same result (Lemma 24.28.4) and hence we conclude

$R\Gamma (U', Lg^*K) = R\Gamma (U, K) \otimes _{\mathcal{A}(U)}^\mathbf {L} \mathcal{A}'(U')$

Finally, let $f' : U' \to V'$ be a morphism in $\mathcal{C}'$ and denote $f = u(f') : U = u(U') \to V = u(V')$ the image in $\mathcal{C}$ . If $K$ is in $\mathit{QC}(\mathcal{A}, \text{d})$ then we have

$\begin{align*} R\Gamma (V', Lg^*K) \otimes _{\mathcal{A}'(V')}^\mathbf {L} \mathcal{A}'(U') & = R\Gamma (V, K) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}'(V') \otimes _{\mathcal{A}'(V')}^\mathbf {L} \mathcal{A}'(U') \\ & = R\Gamma (V, K) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}'(U') \\ & = R\Gamma (V, K) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}(U) \otimes _{\mathcal{A}(U)}^\mathbf {L} \mathcal{A}'(U') \\ & = R\Gamma (U, K) \otimes _{\mathcal{A}(U)}^\mathbf {L} \mathcal{A}'(U') \\ & = R\Gamma (U', Lg^*K) \end{align*}$

as desired. Here we have used the observation above both for $U'$ and $V'$ . $\square$

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24.33 Differential graded modules on a category

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