The Stacks project

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:

  1. any contravariant cohomological functor $H : \mathit{QC}(\mathcal{A}, \text{d}) \to \textit{Ab}$ which transforms direct sums into products is representable,

  2. 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

  3. 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 $V'$. $\square$

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 0GZ8. Beware of the difference between the letter 'O' and the digit '0'.