The Stacks project

Lemma 24.24.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The forgetful functor $F : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A})$ has a left adjoint $G : \textit{Mod}(\mathcal{A}) \to \textit{Mod}(\mathcal{A}, \text{d})$.

Proof. To prove the existence of $G$ we can use the adjoint functor theorem, see Categories, Theorem 4.25.3 (observe that we have switched the roles of $F$ and $G$). The exactness conditions on $F$ are satisfied by Lemma 24.13.2. The set theoretic condition can be seen as follows: suppose given a graded $\mathcal{A}$-module $\mathcal{N}$. Then for any map

\[ \varphi : \mathcal{N} \longrightarrow F(\mathcal{M}) \]

we can consider the smallest differential graded $\mathcal{A}$-submodule $\mathcal{M}' \subset \mathcal{M}$ with $\mathop{\mathrm{Im}}(\varphi ) \subset F(\mathcal{M}')$. It is clear that $\mathcal{M}'$ is the image of the map of graded $\mathcal{A}$-modules

\[ \mathcal{N} \oplus \mathcal{N}[-1] \otimes _\mathcal {O} \mathcal{A} \longrightarrow \mathcal{M} \]

defined by

\[ (n, \sum n_ i \otimes a_ i) \longmapsto \varphi (n) + \sum \text{d}(\varphi (n_ i)) a_ i \]

because the image of this map is easily seen to be a differential graded submodule of $\mathcal{M}$. Thus the number of possible isomorphism classes of these $\mathcal{M}'$ is bounded and we conclude. $\square$

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Your email address will not be published.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