Derived Category D(A)

The derived category $D(\mathcal{A})$ is the central construction in Gelfand-Manin's approach to homological algebra. It is obtained by inverting quasi-isomorphisms in the homotopy category, creating a setting where complexes with the same cohomology become isomorphic. Derived functors, spectral sequences, and duality theorems all find their natural home here.

Construction

Definition6.1Derived Category

The derived category $D(\mathcal{A})$ of an abelian category $\mathcal{A}$ is the localization

$D(\mathcal{A}) = K(\mathcal{A})[\mathrm{qis}^{-1}]$

of the homotopy category at the class of quasi-isomorphisms. It is a triangulated category by Verdier localization.

Equivalently, $D(\mathcal{A}) = K(\mathcal{A}) / K_{\mathrm{ac}}(\mathcal{A})$ , the Verdier quotient by the thick subcategory of acyclic complexes.

Definition6.2Morphisms in D(A)

Morphisms in $D(\mathcal{A})$ are represented by roofs: $A \xleftarrow{s} C \xrightarrow{f} B$ where $s$ is a quasi-isomorphism. Two roofs are equivalent if they admit a common refinement.

When $\mathcal{A}$ has enough injectives, $\mathrm{Hom}_{D(\mathcal{A})}(A, B) \cong \mathrm{Hom}_{K(\mathcal{A})}(A, I_B^\bullet)$ where $I_B^\bullet$ is an injective resolution of $B$ .

Examples

ExampleD(R-Mod)

For a ring $R$ , $D(R\text{-}\mathbf{Mod})$ is the derived category of $R$ -modules. Objects are complexes of $R$ -modules; morphisms are chain maps modulo homotopy, with quasi-isomorphisms inverted. $\mathrm{Hom}_{D(R)}(M[0], N[n]) = \mathrm{Ext}^n_R(M, N)$ .

ExampleD(Ab)

$D(\mathbf{Ab})$ is the derived category of abelian groups. Since $\mathbb{Z}$ has global dimension 1, every object in $D^b(\mathbf{Ab})$ is isomorphic to a direct sum $\bigoplus_n H^n[{-n}]$ of its cohomology groups (placed in the appropriate degrees).

ExampleD^b(Coh(X))

For a smooth projective variety $X$ , $D^b(\mathrm{Coh}(X))$ is the bounded derived category of coherent sheaves. It is a fundamental invariant of $X$ : the Bondal-Orlov reconstruction theorem states that $X$ can be recovered from $D^b(\mathrm{Coh}(X))$ (for varieties with ample or anti-ample canonical bundle).

ExampleEmbedding of A into D(A)

The functor $\mathcal{A} \to D(\mathcal{A})$ sending $A \mapsto A[0]$ (complex concentrated in degree 0) is exact and fully faithful. It identifies $\mathcal{A}$ with the heart of the standard t-structure on $D(\mathcal{A})$ .

ExampleDerived tensor product

In $D(R)$ , the derived tensor product $A \otimes^L B$ is defined by $P_A \otimes B$ where $P_A \to A$ is a projective resolution. $H^{-n}(A \otimes^L B) = \mathrm{Tor}_n(A, B)$ .

ExampleRHom

$\mathrm{RHom}(A, B) = \mathrm{Hom}(A, I_B)$ where $B \to I_B$ is an injective resolution. $H^n(\mathrm{RHom}(A, B)) = \mathrm{Ext}^n(A, B)$ .

ExampleFourier-Mukai transforms

An equivalence $D^b(\mathrm{Coh}(X)) \simeq D^b(\mathrm{Coh}(Y))$ is called a Fourier-Mukai equivalence. By the Orlov representability theorem, every exact equivalence between derived categories of smooth projective varieties is of Fourier-Mukai type.

ExampleTilting

A tilting object $T$ in $D^b(\mathcal{A})$ gives an equivalence $D^b(\mathcal{A}) \simeq D^b(\mathrm{mod}\text{-}\mathrm{End}(T))$ . This is the derived Morita theory.

ExampleSerre duality in D^b

For a smooth projective variety $X$ of dimension $n$ , Serre duality gives $\mathrm{Hom}_{D^b}(A, B) \cong \mathrm{Hom}_{D^b}(B, A \otimes \omega_X[n])^*$ . The Serre functor is $S(-) = (-) \otimes \omega_X[n]$ .

ExampleDerived categories of an elliptic curve

For an elliptic curve $E$ , $D^b(\mathrm{Coh}(E)) \simeq D^b(\mathrm{Coh}(\hat{E}))$ where $\hat{E}$ is the dual elliptic curve. This is the original Fourier-Mukai equivalence, generalizing the Fourier transform.

ExampleD(A) vs K(A)

In $D(\mathcal{A})$ , quasi-isomorphic complexes are isomorphic. In $K(\mathcal{A})$ , only homotopy equivalent complexes are isomorphic. The passage $K \to D$ identifies more objects: a module and its projective resolution become isomorphic in $D$ but not in $K$ .

ExampleExceptional collections

An exceptional collection $\langle E_1, \ldots, E_n \rangle$ in $D^b(\mathrm{Coh}(X))$ generates the derived category if every object is built from $E_i$ by shifts, cones, and direct sums. For $\mathbb{P}^n$ , the exceptional collection $\langle \mathcal{O}, \mathcal{O}(1), \ldots, \mathcal{O}(n) \rangle$ generates.

Key Properties

Theorem6.1Fundamental properties of D(A)

$D(\mathcal{A})$ is a triangulated category.
$H^n : D(\mathcal{A}) \to \mathcal{A}$ is a cohomological functor.
$\mathrm{Hom}_{D(\mathcal{A})}(A[0], B[n]) = \mathrm{Ext}^n_{\mathcal{A}}(A, B)$ for objects $A, B \in \mathcal{A}$ .
If $\mathcal{A}$ has enough injectives, $D^+(\mathcal{A}) \simeq K^+(\mathrm{Inj}\, \mathcal{A})$ .
If $\mathcal{A}$ has enough projectives, $D^-(\mathcal{A}) \simeq K^-(\mathrm{Proj}\, \mathcal{A})$ .

RemarkThe derived category philosophy

The derived category embodies the principle that "complexes are more natural than individual objects." In Gelfand-Manin's approach, derived functors, spectral sequences, and duality are all expressed most cleanly as functors between derived categories, rather than as sequences of functors $R^nF$ on individual objects.