## 24.26 The derived category

This section is the analogue of Differential Graded Algebra, Section 22.22.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. We will construct the derived category $D(\mathcal{A}, \text{d})$ by inverting the quasi-isomorphisms in $K(\textit{Mod}(\mathcal{A}, \text{d}))$.

Lemma 24.26.1. 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 functor $H^0 : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{O})$ of Section 24.13 factors through a functor

$H^0 : K(\textit{Mod}(\mathcal{A}, \text{d})) \to \textit{Mod}(\mathcal{O})$

which is homological in the sense of Derived Categories, Definition 13.3.5.

Proof. It follows immediately from the definitions that there is a commutative diagram

$\xymatrix{ \textit{Mod}(\mathcal{A}, \text{d}) \ar[r] \ar[d] & K(\textit{Mod}(\mathcal{A}, \text{d})) \ar[d] \\ \text{Comp}(\mathcal{O}) \ar[r] & K(\textit{Mod}(\mathcal{O})) }$

Since $H^0(\mathcal{M})$ is defined as the zeroth cohomology sheaf of the underlying complex of $\mathcal{O}$-modules of $\mathcal{M}$ the lemma follows from the case of complexes of $\mathcal{O}$-modules which is a special case of Derived Categories, Lemma 13.11.1. $\square$

Lemma 24.26.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The full subcategory $\text{Ac}$ of the homotopy category $K(\textit{Mod}(\mathcal{A}, \text{d}))$ consisting of acyclic modules is a strictly full saturated triangulated subcategory of $K(\textit{Mod}(\mathcal{A}, \text{d}))$.

Proof. Of course an object $\mathcal{M}$ of $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is in $\text{Ac}$ if and only if $H^ i(\mathcal{M}) = H^0(\mathcal{M}[i])$ is zero for all $i$. The lemma follows from this, Lemma 24.26.1, and Derived Categories, Lemma 13.6.3. See also Derived Categories, Definitions 13.6.1 and 13.3.4 and Lemma 13.4.16. $\square$

Lemma 24.26.3. 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 subclass $\text{Qis} \subset \text{Arrows}(K(\textit{Mod}(\mathcal{A}, \text{d})))$ consisting of quasi-isomorphisms. This is a saturated multiplicative system compatible with the triangulated structure on $K(\textit{Mod}(\mathcal{A}, \text{d}))$.

Proof. Observe that if $f , g : \mathcal{M} \to \mathcal{N}$ are morphisms of $\textit{Mod}(\mathcal{A}, \text{d})$ which are homotopic, then $f$ is a quasi-isomorphism if and only if $g$ is a quasi-isomorphism. Namely, the maps $H^ i(f) = H^0(f[i])$ and $H^ i(g) = H^0(g[i])$ are the same by Lemma 24.26.1. Thus it is unambiguous to say that a morphism of the homotopy category $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is a quasi-isomorphism. For definitions of “multiplicative system”, “saturated”, and “compatible with the triangulated structure” see Derived Categories, Definition 13.5.1 and Categories, Definitions 4.27.1 and 4.27.20.

To actually prove the lemma consider the composition of exact functors of triangulated categories

$K(\textit{Mod}(\mathcal{A}, \text{d})) \longrightarrow K(\textit{Mod}(\mathcal{O})) \longrightarrow D(\mathcal{O})$

and observe that a morphism $f : \mathcal{M} \to \mathcal{N}$ of $K(\textit{Mod}(\mathcal{A}, \text{d}))$ is in $\text{Qis}$ if and only if it maps to an isomorphism in $D(\mathcal{O})$. Thus the lemma follows from Derived Categories, Lemma 13.5.3. $\square$

In the situation of Lemma 24.26.3 we can apply Derived Categories, Proposition 13.5.5 to obtain an exact functor of triangulated categories

$Q : K(\textit{Mod}(\mathcal{A}, \text{d})) \longrightarrow \text{Qis}^{-1}K(\textit{Mod}(\mathcal{A}, \text{d}))$

However, as $\textit{Mod}(\mathcal{A}, \text{d})$ is a “big” category, i.e., its objects form a proper class, it isn't immediately clear that given $\mathcal{M}$ and $\mathcal{N}$ the construction of $\text{Qis}^{-1}K(\textit{Mod}(\mathcal{A}, \text{d}))$ produces a set

$\mathop{\mathrm{Mor}}\nolimits _{\text{Qis}^{-1}K(\textit{Mod}(\mathcal{A}, \text{d}))} (\mathcal{M}, \mathcal{N})$

of morphisms. However, this is true thanks to our construction of K-injective complexes. Namely, by Theorem 24.25.13 we can choose a quasi-isomorphism $s_0 : \mathcal{N} \to \mathcal{I}$ where $\mathcal{I}$ is a graded injective and K-injective differential graded $\mathcal{A}$-module. Next, recall that elements of the displayed set are equivalence classes of pairs $(f : \mathcal{M} \to \mathcal{N}', s : \mathcal{N} \to \mathcal{N}')$ where $f$ is an arbitrary morphism of $K(\textit{Mod}(\mathcal{A}, \text{d}))$ and $s$ is a quasi-isomorphsm, see the description of the left calculus of fractions in Categories, Section 4.27. By Lemma 24.25.10 we can choose the dotted arrow

$\xymatrix{ \mathcal{M} \ar[rd]^ f & & \mathcal{N} \ar[ld]_ s \ar[rd]^{s_0} \\ & \mathcal{N}' \ar@{..>}[rr]^{s'} & & \mathcal{I} }$

making the diagram commute (in the homotopy category). Thus the pair $(f, s)$ is equivalent to the pair $(s' \circ f, s_0)$ and we find that the collection of equivalence classes forms a set.

Definition 24.26.4. 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 $\text{Qis}$ be as in Lemma 24.26.3. The derived category of $(\mathcal{A}, \text{d})$ is the triangulated category

$D(\mathcal{A}, \text{d}) = \text{Qis}^{-1}K(\textit{Mod}(\mathcal{A}, \text{d}))$

discussed in more detail above.

Lemma 24.26.5. In Definition 24.26.4 the kernel of the localization functor $Q : K(\textit{Mod}(\mathcal{A}, \text{d})) \to D(\mathcal{A}, \text{d})$ is the category $\text{Ac}$ of Lemma 24.26.2.

Proof. This is immediate from Derived Categories, Lemma 13.5.8 and the fact that $0 \to \mathcal{M}$ is a quasi-isomorphism if and only if $\mathcal{M}$ is acyclic. $\square$

Lemma 24.26.6. In Definition 24.26.4 the functor $H^0 : K(\textit{Mod}(\mathcal{A}, \text{d})) \to \textit{Mod}(\mathcal{O})$ factors through a homological functor $H^0 : D(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{O})$.

Proof. Follows immediately from Derived Categories, Lemma 13.5.6. $\square$

Here is the promised lemma computing morphism sets in the derived category.

Lemma 24.26.7. 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 $\mathcal{M}$ and $\mathcal{N}$ be differential graded $\mathcal{A}$-modules. Let $\mathcal{N} \to \mathcal{I}$ be a quasi-isomorphism with $\mathcal{I}$ a graded injective and K-injective differential graded $\mathcal{A}$-module. Then

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}, \mathcal{I})$

Proof. Since $\mathcal{N} \to \mathcal{I}$ is a quasi-isomorphism we see that

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{N}) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{I})$

In the discussion preceding Definition 24.26.4 we found, using Lemma 24.25.10, that any morphism $\mathcal{M} \to \mathcal{I}$ in $D(\mathcal{A}, \text{d})$ can be represented by a morphism $f : \mathcal{M} \to \mathcal{I}$ in $K(\textit{Mod}(\mathcal{A}, \text{d}))$. Now, if $f, f' : \mathcal{M} \to \mathcal{I}$ are two morphism in $K(\textit{Mod}(\mathcal{A}, \text{d}))$, then they define the same morphism in $D(\mathcal{A}, \text{d})$ if and only if there exists a quasi-isomorphism $g : \mathcal{I} \to \mathcal{K}$ in $K(\textit{Mod}(\mathcal{A}, \text{d}))$ such that $g \circ f = g \circ f'$, see Categories, Lemma 4.27.6. However, by Lemma 24.25.10 there exists a map $h : \mathcal{K} \to \mathcal{I}$ such that $h \circ g = \text{id}_\mathcal {I}$ in in $K(\textit{Mod}(\mathcal{A}, \text{d}))$. Thus $g \circ f = g \circ f'$ implies $f = f'$ and the proof is complete. $\square$

Lemma 24.26.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Then

1. $D(\mathcal{A}, \text{d})$ has both direct sums and products,

2. direct sums are obtained by taking direct sums of differential graded $\mathcal{A}$-modules,

3. products are obtained by taking products of K-injective differential graded modules.

Proof. We will use that $\textit{Mod}(\mathcal{A}, \text{d})$ is an abelian category with arbitrary direct sums and products, and that these give rise to direct sums and products in $K(\textit{Mod}(\mathcal{A}, \text{d}))$. See Lemmas 24.13.2 and 24.21.3.

Let $\mathcal{M}_ j$ be a family of differential graded $\mathcal{A}$-modules. Consider the direct sum $\mathcal{M} = \bigoplus \mathcal{M}_ j$ as a differential graded $\mathcal{A}$-module. For a differential graded $\mathcal{A}$-module $\mathcal{N}$ choose a quasi-isomorphism $\mathcal{N} \to \mathcal{I}$ where $\mathcal{I}$ is graded injective and K-injective as a differential graded $\mathcal{A}$-module. See Theorem 24.25.13. Using Lemma 24.26.7 we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{N}) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{I}) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A}, \text{d})}(\mathcal{M}_ j, \mathcal{I}) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}_ j, \mathcal{I}) \end{align*}

whence the existence of direct sums in $D(A, \text{d})$ as given in part (2) of the lemma.

Let $\mathcal{M}_ j$ be a family of differential graded $\mathcal{A}$-modules. For each $j$ choose a quasi-isomorphism $\mathcal{M} \to \mathcal{I}_ j$ where $\mathcal{I}_ j$ is graded injective and K-injective as a differential graded $\mathcal{A}$-module. Consider the product $\mathcal{I} = \prod \mathcal{I}_ j$ of differential graded $\mathcal{A}$-modules. By Lemmas 24.25.8 and 24.25.4 we see that $\mathcal{I}$ is graded injective and K-injective as a differential graded $\mathcal{A}$-module. For a differential graded $\mathcal{A}$-module $\mathcal{N}$ using Lemma 24.26.7 we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{I}) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{I}) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{I}_ j) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{M}_ j) \end{align*}

whence the existence of products in $D(\mathcal{A}, \text{d})$ as given in part (3) of the lemma. $\square$

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