The Stacks project

22.22 The derived category

Recall that the notions of acyclic differential graded modules and quasi-isomorphism of differential graded modules make sense (see Section 22.4).

Lemma 22.22.1. Let $(A, \text{d})$ be a differential graded algebra. The full subcategory $\text{Ac}$ of $K(\text{Mod}_{(A, \text{d})})$ consisting of acyclic modules is a strictly full saturated triangulated subcategory of $K(\text{Mod}_{(A, \text{d})})$. The corresponding saturated multiplicative system (see Derived Categories, Lemma 13.6.10) of $K(\text{Mod}_{(A, \text{d})})$ is the class $\text{Qis}$ of quasi-isomorphisms. In particular, the kernel of the localization functor

\[ Q : K(\text{Mod}_{(A, \text{d})}) \to \text{Qis}^{-1}K(\text{Mod}_{(A, \text{d})}) \]

is $\text{Ac}$. Moreover, the functor $H^0$ factors through $Q$.

Proof. We know that $H^0$ is a homological functor by the long exact sequence of homology ( The kernel of $H^0$ is the subcategory of acyclic objects and the arrows with induce isomorphisms on all $H^ i$ are the quasi-isomorphisms. Thus this lemma is a special case of Derived Categories, Lemma 13.6.11.

Set theoretical remark. The construction of the localization in Derived Categories, Proposition 13.5.5 assumes the given triangulated category is “small”, i.e., that the underlying collection of objects forms a set. Let $V_\alpha $ be a partial universe (as in Sets, Section 3.5) containing $(A, \text{d})$ and where the cofinality of $\alpha $ is bigger than $\aleph _0$ (see Sets, Proposition 3.7.2). Then we can consider the category $\text{Mod}_{(A, \text{d}), \alpha }$ of differential graded $A$-modules contained in $V_\alpha $. A straightforward check shows that all the constructions used in the proof of Proposition 22.10.3 work inside of $\text{Mod}_{(A, \text{d}), \alpha }$ (because at worst we take finite direct sums of differential graded modules). Thus we obtain a triangulated category $\text{Qis}_\alpha ^{-1}K(\text{Mod}_{(A, \text{d}), \alpha })$. We will see below that if $\beta > \alpha $, then the transition functors

\[ \text{Qis}_\alpha ^{-1}K(\text{Mod}_{(A, \text{d}), \alpha }) \longrightarrow \text{Qis}_\beta ^{-1}K(\text{Mod}_{(A, \text{d}), \beta }) \]

are fully faithful as the morphism sets in the quotient categories are computed by maps in the homotopy categories from P-resolutions (the construction of a P-resolution in the proof of Lemma 22.20.4 takes countable direct sums as well as direct sums indexed over subsets of the given module). The reader should therefore think of the category of the lemma as the union of these subcategories. $\square$

Taking into account the set theoretical remark at the end of the proof of the preceding lemma we define the derived category as follows.

Definition 22.22.2. Let $(A, \text{d})$ be a differential graded algebra. Let $\text{Ac}$ and $\text{Qis}$ be as in Lemma 22.22.1. The derived category of $(A, \text{d})$ is the triangulated category

\[ D(A, \text{d}) = K(\text{Mod}_{(A, \text{d})})/\text{Ac} = \text{Qis}^{-1}K(\text{Mod}_{(A, \text{d})}). \]

We denote $H^0 : D(A, \text{d}) \to \text{Mod}_ R$ the unique functor whose composition with the quotient functor gives back the functor $H^0$ defined above.

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

Lemma 22.22.3. Let $(A, \text{d})$ be a differential graded algebra. Let $M$ and $N$ be differential graded $A$-modules.

  1. Let $P \to M$ be a P-resolution as in Lemma 22.20.4. Then

    \[ \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(M, N) = \mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(P, N) \]
  2. Let $N \to I$ be an I-resolution as in Lemma 22.21.4. Then

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

Proof. Let $P \to M$ be as in (1). Since $P \to M$ is a quasi-isomorphism we see that

\[ \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(P, N) = \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(M, N) \]

by definition of the derived category. A morphism $f : P \to N$ in $D(A, \text{d})$ is equal to $s^{-1}f'$ where $f' : P \to N'$ is a morphism and $s : N \to N'$ is a quasi-isomorphism. Choose a distinguished triangle

\[ N \to N' \to Q \to N[1] \]

As $s$ is a quasi-isomorphism, we see that $Q$ is acyclic. Thus $\mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(P, Q[k]) = 0$ for all $k$ by Lemma 22.20.2. Since $\mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(P, -)$ is cohomological, we conclude that we can lift $f' : P \to N'$ uniquely to a morphism $f : P \to N$. This finishes the proof.

The proof of (2) is dual to that of (1) using Lemma 22.21.2 in stead of Lemma 22.20.2. $\square$

Lemma 22.22.4. Let $(A, \text{d})$ be a differential graded algebra. Then

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

  2. direct sums are obtained by taking direct sums of differential graded modules,

  3. products are obtained by taking products of differential graded modules.

Proof. We will use that $\text{Mod}_{(A, \text{d})}$ is an abelian category with arbitrary direct sums and products, and that these give rise to direct sums and products in $K(\text{Mod}_{(A, \text{d})})$. See Lemmas 22.4.2 and 22.5.4.

Let $M_ j$ be a family of differential graded $A$-modules. Consider the graded direct sum $M = \bigoplus M_ j$ which is a differential graded $A$-module with the obvious. For a differential graded $A$-module $N$ choose a quasi-isomorphism $N \to I$ where $I$ is a differential graded $A$-module with property (I). See Lemma 22.21.4. Using Lemma 22.22.3 we have

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

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

Let $M_ j$ be a family of differential graded $A$-modules. Consider the product $M = \prod M_ j$ of differential graded $A$-modules. For a differential graded $A$-module $N$ choose a quasi-isomorphism $P \to N$ where $P$ is a differential graded $A$-module with property (P). See Lemma 22.20.4. Using Lemma 22.22.3 we have

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

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

Remark 22.22.5. Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded $R$-algebra. Using P-resolutions we can sometimes reduce statements about general objects of $D(A, \text{d})$ to statements about $A[k]$. Namely, let $T$ be a property of objects of $D(A, \text{d})$ and assume that

  1. if $K_ i$, $i \in I$ is a family of objects of $D(A, \text{d})$ and $T(K_ i)$ holds for all $i \in I$, then $T(\bigoplus K_ i)$,

  2. if $K \to L \to M \to K[1]$ is a distinguished triangle of $D(A, \text{d})$ and $T$ holds for two, then $T$ holds for the third object, and

  3. $T(A[k])$ holds for all $k \in \mathbf{Z}$.

Then $T$ holds for all objects of $D(A, \text{d})$. This is clear from Lemmas 22.20.1 and 22.20.4.

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