## 24.27 The canonical delta-functor

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Consider the functor $\textit{Mod}(\mathcal{A}, \text{d}) \to K(\textit{Mod}(\mathcal{A}, \text{d}))$. This functor is not a $\delta$-functor in general. However, it turns out that the functor $\textit{Mod}(\mathcal{A}, \text{d}) \to D(A, \text{d})$ is a $\delta$-functor. In order to see this we have to define the morphisms $\delta$ associated to a short exact sequence

$0 \to \mathcal{K} \xrightarrow {a} \mathcal{L} \xrightarrow {b} \mathcal{M} \to 0$

in the abelian category $\textit{Mod}(\mathcal{A}, \text{d})$. Consider the cone $C(a)$ of the morphism $a$ together with its canonical morphisms $i : \mathcal{L} \to C(a)$ and $p : C(a) \to \mathcal{K}[1]$, see Definition 24.22.2. There is a homomorphism of differential graded $\mathcal{A}$-modules

$q : C(a) \longrightarrow \mathcal{M}$

by Differential Graded Algebra, Lemma 22.27.3 (which we may use by the discussion in Section 24.22) applied to the diagram

$\xymatrix{ \mathcal{K} \ar[r]_ a \ar[d] & \mathcal{L} \ar[d]^ b \\ 0 \ar[r] & \mathcal{M} }$

The map $q$ is a quasi-isomorphism for example because this is true in the category of morphisms of complexes of $\mathcal{O}$-modules, see discussion in Derived Categories, Section 13.12. According to Differential Graded Algebra, Lemma 22.27.13 (which we may use by the discussion in Section 24.22) the triangle

$(\mathcal{K}, \mathcal{L}, C(a), a, i, -p)$

is a distinguished triangle in $K(\textit{Mod}(\mathcal{A}, \text{d}))$. As the localization functor $K(\textit{Mod}(\mathcal{A}, \text{d})) \to D(\mathcal{A}, \text{d})$ is exact we see that $(\mathcal{K}, \mathcal{L}, C(a), a, i, -p)$ is a distinguished triangle in $D(\mathcal{A}, \text{d})$. Since $q$ is a quasi-isomorphism we see that $q$ is an isomorphism in $D(\mathcal{A}, \text{d})$. Hence we deduce that

$(\mathcal{K}, \mathcal{L}, \mathcal{M}, a, b, -p \circ q^{-1})$

is a distinguished triangle of $D(\mathcal{A}, \text{d})$. This suggests the following lemma.

Lemma 24.27.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The localization functor $\textit{Mod}(\mathcal{A}, \text{d}) \to D(\mathcal{A}, \text{d})$ has the natural structure of a $\delta$-functor, with

$\delta _{\mathcal{K} \to \mathcal{L} \to \mathcal{M}} = - p \circ q^{-1}$

with $p$ and $q$ as explained above.

Proof. We have already seen that this choice leads to a distinguished triangle whenever given a short exact sequence of complexes. We have to show functoriality of this construction, see Derived Categories, Definition 13.3.6. This follows from Differential Graded Algebra, Lemma 22.27.3 (which we may use by the discussion in Section 24.22) with a bit of work. Compare with Derived Categories, Lemma 13.12.1. $\square$

Lemma 24.27.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}_ n$ be a system of differential graded $\mathcal{A}$-modules. Then the derived colimit $\text{hocolim} \mathcal{M}_ n$ in $D(\mathcal{A}, \text{d})$ is represented by the differential graded module $\mathop{\mathrm{colim}}\nolimits \mathcal{M}_ n$.

Proof. Set $\mathcal{M} = \mathop{\mathrm{colim}}\nolimits \mathcal{M}_ n$. We have an exact sequence of differential graded $\mathcal{A}$-modules

$0 \to \bigoplus \mathcal{M}_ n \to \bigoplus \mathcal{M}_ n \to \mathcal{M} \to 0$

by Derived Categories, Lemma 13.33.6 (applied the underlying complexes of $\mathcal{O}$-modules). The direct sums are direct sums in $D(\mathcal{A}, \text{d})$ by Lemma 24.26.8. Thus the result follows from the definition of derived colimits in Derived Categories, Definition 13.33.1 and the fact that a short exact sequence of complexes gives a distinguished triangle (Lemma 24.27.1). $\square$

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