The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

22.16 The canonical delta-functor

Let $(A, \text{d})$ be a differential graded algebra. Consider the functor $\text{Mod}_{(A, \text{d})} \to K(\text{Mod}_{(A, \text{d})})$. This functor is not a $\delta $-functor in general. However, it turns out that the functor $\text{Mod}_{(A, \text{d})} \to D(A, \text{d})$ is a $\delta $-functor. In order to see this we have to define the morphisms $\delta $ associated to a short exact sequence

\[ 0 \to K \xrightarrow {a} L \xrightarrow {b} M \to 0 \]

in the abelian category $\text{Mod}_{(A, \text{d})}$. Consider the cone $C(a)$ of the morphism $a$. We have $C(a) = L \oplus K$ and we define $q : C(a) \to M$ via the projection to $L$ followed by $b$. Hence a homomorphism of differential graded $A$-modules

\[ q : C(a) \longrightarrow M. \]

It is clear that $q \circ i = b$ where $i$ is as in Definition 22.6.1. Note that, as $a$ is injective, the kernel of $q$ is identified with the cone of $\text{id}_ K$ which is acyclic. Hence we see that $q$ is a quasi-isomorphism. According to Lemma 22.9.4 the triangle

\[ (K, L, C(a), a, i, -p) \]

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

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

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

Lemma 22.16.1. Let $(A, \text{d})$ be a differential graded algebra. The functor $\text{Mod}_{(A, \text{d})} \to D(A, \text{d})$ defined has the natural structure of a $\delta $-functor, with

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

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 Lemma 22.6.2 with a bit of work. Compare with Derived Categories, Lemma 13.12.1. $\square$

Lemma 22.16.2. Let $(A, \text{d})$ be a differential graded algebra. Let $M_ n$ be a system of differential graded modules. Then the derived colimit $\text{hocolim} M_ n$ in $D(A, \text{d})$ is represented by the differential graded module $\mathop{\mathrm{colim}}\nolimits M_ n$.

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

\[ 0 \to \bigoplus M_ n \to \bigoplus M_ n \to M \to 0 \]

by Derived Categories, Lemma 13.31.6 (applied the underlying complexes of abelian groups). The direct sums are direct sums in $D(\mathcal{A})$ by Lemma 22.15.4. Thus the result follows from the definition of derived colimits in Derived Categories, Definition 13.31.1 and the fact that a short exact sequence of complexes gives a distinguished triangle (Lemma 22.16.1). $\square$


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