Lemma 24.22.3. 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 differential graded category $\textit{Mod}^{dg}(\mathcal{A}, \text{d})$ satisfies axiom (C) formulated in Differential Graded Algebra, Situation 22.27.2.
Proof. Let $f : \mathcal{K} \to \mathcal{L}$ be a homomorphism of differential graded $\mathcal{A}$-modules. By the above we have an admissible short exact sequence
\[ 0 \to \mathcal{L} \to C(f) \to \mathcal{K}[1] \to 0 \]
To finish the proof we have to show that the boundary map
\[ \delta : \mathcal{K}[1] \to \mathcal{L}[1] \]
associated to this (see discussion above) is equal to $f[1]$. For the section $s : \mathcal{K}[1] \to C(f)$ we use in degree $n$ the embedding $\mathcal{K}^{n + 1} \to C(f)^ n$. Then in degree $n$ the map $\pi $ is given by the projections $C(f)^ n \to \mathcal{L}^ n$. Then finally we have to compute
\[ \delta = \pi \circ \text{d}_{C(f)} \circ s \]
(see discussion above). In matrix notation this is equal to
\[ \left( \begin{matrix} 1
& 0
\end{matrix} \right) \left( \begin{matrix} \text{d}_\mathcal {L}
& f
\\ 0
& -\text{d}_\mathcal {K}
\end{matrix} \right) \left( \begin{matrix} 0
\\ 1
\end{matrix} \right) = f \]
as desired. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)