## 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.

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.

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

$\mathcal{K} \to \mathcal{L} \to \mathcal{N} \to \mathcal{K}$

where the arrow $\mathcal{N} \to \mathcal{K}$ 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:

1. 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

$d_{C(f)} = \left( \begin{matrix} \text{d}_\mathcal {L} & f \\ 0 & -\text{d}_\mathcal {K} \end{matrix} \right)$
2. the multiplication map

$C(f)^ n \times \mathcal{A}^ m \to C(f)^{n + m}$

is the direct sum of the multiplication map $\mathcal{L}^ n \times \mathcal{A}^ m \to \mathcal{L}^{n + m}$ and the multiplication map $\mathcal{K}^{n + 1} \times \mathcal{A}^ m \to \mathcal{K}^{n + 1 + m}$.

It comes equipped with canonical hommorphisms of differential graded $\mathcal{A}$-modules $i : \mathcal{L} \to C(f)$ and $p : C(f) \to \mathcal{K}$ induced by the obvious maps.

Observe that in the situation of the definition the sequence

$0 \to \mathcal{L} \to C(f) \to \mathcal{K} \to 0$

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

$0 \to \mathcal{L} \to C(f) \to \mathcal{K} \to 0$

To finish the proof we have to show that the boundary map

$\delta : \mathcal{K} \to \mathcal{L}$

associated to this (see discussion above) is equal to $f$. For the section $s : \mathcal{K} \to C(f)$ 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

1. the shift functors are those constructed in Section 24.20,

2. the distinghuished triangles are those triangles in $K(\textit{Mod}(\mathcal{A}, \text{d}))$ which are isomorphic as a triangle to a triangle

$\mathcal{K} \to \mathcal{L} \to \mathcal{N} \xrightarrow {\delta } \mathcal{K},\quad \quad \delta = \pi \circ \text{d}_\mathcal {L} \circ s$

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.

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

$\mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A}, \text{d})}(C, \mathcal{M}) = \{ (x, y) \in \Gamma (\mathcal{C}, \mathcal{M}^0) \times \Gamma (\mathcal{C}, \mathcal{M}^{-1}) \mid x = \text{d}(y)\}$

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 $C^0 = \mathcal{A}^0 \oplus \mathcal{A}^1$.

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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