Lemma 24.22.1. 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 differential graded category $\textit{Mod}^{dg}(\mathcal{A}, \text{d})$ satisfies axioms (A) and (B) of Differential Graded Algebra, Section 22.27.
24.22 Cones and triangles
In this section we use the material from Differential Graded Algebra, Section 22.27 to conclude that the homotopy category of the category of differential graded $\mathcal{A}$-modules is a triangulated category.
Proof. Suppose given differential graded $\mathcal{A}$-modules $\mathcal{M}$ and $\mathcal{N}$. Consider the differential graded $\mathcal{A}$-module $\mathcal{M} \oplus \mathcal{N}$ defined in the obvious manner. Then the coprojections $i : \mathcal{M} \to \mathcal{M} \oplus \mathcal{N}$ and $j : \mathcal{N} \to \mathcal{M} \oplus \mathcal{N}$ and the projections $p : \mathcal{M} \oplus \mathcal{N} \to \mathcal{N}$ and $q : \mathcal{M} \oplus \mathcal{N} \to \mathcal{M}$ are morphisms of differential graded $\mathcal{A}$-modules. Hence $i, j, p, q$ are homogeneous of degree $0$ and closed, i.e., $\text{d}(i) = 0$, etc. Thus this direct sum is a differential graded sum in the sense of Differential Graded Algebra, Definition 22.26.4. This proves axiom (A).
Axiom (B) was shown in Section 24.20. $\square$
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Recall that a sequence
\[ 0 \to \mathcal{K} \to \mathcal{L} \to \mathcal{N} \to 0 \]
in $\textit{Mod}(\mathcal{A}, \text{d})$ is called an admissible short exact sequence (in Differential Graded Algebra, Section 22.27) if it is split in $\textit{Mod}(\mathcal{A})$. In other words, if it is split as a sequence of graded $\mathcal{A}$-modules. Denote $s : \mathcal{N} \to \mathcal{L}$ and $\pi : \mathcal{L} \to \mathcal{K}$ graded $\mathcal{A}$-module splittings. Combining Lemma 24.22.1 and Differential Graded Algebra, Lemma 22.27.1 we obtain a triangle
\[ \mathcal{K} \to \mathcal{L} \to \mathcal{N} \to \mathcal{K}[1] \]
where the arrow $\mathcal{N} \to \mathcal{K}[1]$ in the proof of Differential Graded Algebra, Lemma 22.27.1 is constructed as
\[ \delta = \pi \circ \text{d}_{\mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})} (\mathcal{L}, \mathcal{M})}(s) = \pi \circ \text{d}_\mathcal {L} \circ s - \pi \circ s \circ \text{d}_\mathcal {N} = \pi \circ \text{d}_\mathcal {L} \circ s \]
with apologies for the horrendous notation. In any case, we see that in our setting the boundary map $\delta $ as constructed in Differential Graded Algebra, Lemma 22.27.1 agrees on underlying complexes of $\mathcal{O}$-modules with the usual boundary map used throughout the Stacks project for termwise split short exact sequences of complexes, see Derived Categories, Definition 13.9.9.
Definition 24.22.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $f : \mathcal{K} \to \mathcal{L}$ be a homomorphism of differential graded $\mathcal{A}$-modules. The cone of $f$ is the differential graded $\mathcal{A}$-module $C(f)$ defined as follows:
the underlying complex of $\mathcal{O}$-modules is the cone of the corresponding map $f : \mathcal{K}^\bullet \to \mathcal{L}^\bullet $ of complexes of $\mathcal{A}$-modules, i.e., we have $C(f)^ n = \mathcal{L}^ n \oplus \mathcal{K}^{n + 1}$ and differential
the multiplication map
is the direct sum of the multiplication map $\mathcal{L}^ n \times \mathcal{A}^ m \to \mathcal{L}^{n + m}$ and the multiplication map $\mathcal{K}^{n + 1} \times \mathcal{A}^ m \to \mathcal{K}^{n + 1 + m}$.
\[ d_{C(f)} = \left( \begin{matrix} \text{d}_\mathcal {L}
& f
\\ 0
& -\text{d}_\mathcal {K}
\end{matrix} \right) \]
\[ C(f)^ n \times \mathcal{A}^ m \to C(f)^{n + m} \]
It comes equipped with canonical hommorphisms of differential graded $\mathcal{A}$-modules $i : \mathcal{L} \to C(f)$ and $p : C(f) \to \mathcal{K}[1]$ induced by the obvious maps.
Observe that in the situation of the definition the sequence
\[ 0 \to \mathcal{L} \to C(f) \to \mathcal{K}[1] \to 0 \]
is an addmissible short exact sequence.
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 embeddding $\mathcal{K}^{n + 1} \to C(f)^ n$. 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. $\square$
At this point we know that all lemmas proved in Differential Graded Algebra, Section 22.27 are valid for the differential graded category $\textit{Mod}^{dg}(\mathcal{A}, \text{d})$. In particular, we have the following.
Proposition 24.22.4. 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 homotopy category $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is a triangulated category where
the shift functors are those constructed in Section 24.20,
the distinghuished triangles are those triangles in $K(\textit{Mod}(\mathcal{A}, \text{d}))$ which are isomorphic as a triangle to a triangle
constructed from an admissible short exact sequence $0 \to \mathcal{K} \to \mathcal{L} \to \mathcal{N} \to 0$ in $\textit{Mod}(\mathcal{A}, \text{d})$ above.
\[ \mathcal{K} \to \mathcal{L} \to \mathcal{N} \xrightarrow {\delta } \mathcal{K}[1],\quad \quad \delta = \pi \circ \text{d}_\mathcal {L} \circ s \]
Proof. Recall that $K(\textit{Mod}(\mathcal{A}, \text{d})) = K(\textit{Mod}^{dg}(\mathcal{A}, \text{d}))$, see Section 24.21. Having said this, the proposition follows from Lemmas 24.22.1 and 24.22.3 and Differential Graded Algebra, Proposition 22.27.16. $\square$
Remark 24.22.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $C = C(\text{id}_\mathcal {A})$ be the cone on the identity map $\mathcal{A} \to \mathcal{A}$ viewed as a map of differential graded $\mathcal{A}$-modules. Then where the map from left to right sends $f$ to the pair $(x, y)$ where $x$ is the image of the global section $(0, 1)$ of $C^{-1} = \mathcal{A}^{-1} \oplus \mathcal{A}^0$ and where $y$ is the image of the global section $(1, 0)$ of $C^0 = \mathcal{A}^0 \oplus \mathcal{A}^1$.
\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A}, \text{d})}(C, \mathcal{M}) = \{ (x, y) \in \Gamma (\mathcal{C}, \mathcal{M}^0) \times \Gamma (\mathcal{C}, \mathcal{M}^{-1}) \mid x = \text{d}(y)\} \]
Lemma 24.22.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a differential graded $\mathcal{O}$-algebra. The category $\textit{Mod}(\mathcal{A}, \text{d})$ is a Grothendieck abelian category.
Proof. By Lemma 24.13.2 and the definition of a Grothendieck abelian category (Injectives, Definition 19.10.1) it suffices to show that $\textit{Mod}(\mathcal{A}, \text{d})$ has a generator. For every object $U$ of $\mathcal{C}$ we denote $C_ U$ the cone on the identity map $\mathcal{A}_ U \to \mathcal{A}_ U$ as in Remark 24.22.5. We claim that
\[ \mathcal{G} = \bigoplus \nolimits _{k, U} j_{U!}C_ U[k] \]
is a generator where the sum is over all objects $U$ of $\mathcal{C}$ and $k \in \mathbf{Z}$. Indeed, given a differential graded $\mathcal{A}$-module $\mathcal{M}$ if there are no nonzero maps from $\mathcal{G}$ to $\mathcal{M}$, then we see that for all $k$ and $U$ we have
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A})}(j_{U!}C_ U[k], \mathcal{M}) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A}_ U)}(C_ U[k], \mathcal{M}|_ U) \\ & = \{ (x, y) \in \mathcal{M}^{-k}(U) \times \mathcal{M}^{-k - 1}(U) \mid x = \text{d}(y)\} \end{align*}
is equal to zero. Hence $\mathcal{M}$ is zero. $\square$
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