Derived Categories — Mathematics Knowledge System

Overview

The derived category D(A) of an abelian category A is constructed by formally inverting quasi-isomorphisms (chain maps inducing isomorphisms on all cohomology groups). Derived categories provide the natural setting for homological algebra: derived functors (Ext, Tor, RHom, L\otimes) become honest functors between derived categories rather than sequences of cohomology groups. They are indispensable in modern algebraic geometry, representation theory, and mathematical physics.

Intuition

Working in an abelian category, long exact sequences of cohomology are cumbersome. The derived category packages all cohomological information into a single object: a complex. Two complexes are identified (via quasi-isomorphism) whenever they have the same cohomology in every degree. Derived functors then become exact in the strongest sense — they preserve distinguished triangles (the derived-category replacement for short exact sequences).

Formal Definition

Definition

Given an abelian category A, the derived category D(A) is the localization of the category of chain complexes Ch(A) at the class of quasi-isomorphisms. It is constructed in two steps: first form the homotopy category K(A) (identify homotopic chain maps), then invert quasi-isomorphisms.

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

Derived category as localization of the homotopy category at quasi-isomorphisms

derived-category

\mathrm{Hom}_{D(A)}(X, Y) = \varinjlim_{X \xrightarrow{\sim} X'} \mathrm{Hom}_{K(A)}(X', Y)

Morphisms in D(A) are roofs: spans where the left leg is a quasi-isomorphism

hom-in-derived

X \to Y \to Z \xrightarrow{[1]} X[1]

Distinguished triangles replace short exact sequences; [1] denotes the shift functor

distinguished-triangle

RF: D^+(A) \to D^+(B), \quad RF(X) = F(I^\bullet) \text{ where } X \xrightarrow{\sim} I^\bullet \text{ injective resolution}

Right derived functor RF of a left-exact functor F: A -> B

right-derived-functor

Notation

Notation	Meaning
$D(A)$	Derived category of abelian category A
$D^+(A)$	Derived category of complexes bounded below
$D^b(A)$	Bounded derived category
$X[n]$	Shift (translation) of complex X by n
$H^n(X)$	n-th cohomology object of complex X
$R\mathrm{Hom}(-, -)$	Derived internal hom
$(-) \otimes^L (-)$	Left derived tensor product
$Rf_*$	Right derived pushforward
$Lf^*$	Left derived pullback

Theorems

Theorem 1: Existence of Derived Functors

If A has enough injectives and F: A \to B is left-exact, then RF: D^+(A) \to D^+(B) exists and H^n(RF(X)) \cong R^nF(H^0(X)) when X is concentrated in degree 0.

Theorem 2: Grothendieck Duality

For a proper morphism f: X \to Y of Noetherian schemes, there is a right adjoint f^!: D^+_{\mathrm{qcoh}}(Y) \to D^+_{\mathrm{qcoh}}(X) to Rf_*: Rf_* \dashv f^!, generalizing Serre duality.

Theorem 3: Bondal-Orlov Reconstruction

Let X be a smooth projective variety with ample (or anti-ample) canonical bundle. If D^b(\mathrm{Coh}(X)) \cong D^b(\mathrm{Coh}(Y)) as triangulated categories, then X \cong Y.

Worked Examples

Choose an injective resolution 0 -> N -> I^0 -> I^1 -> ... of N. In the derived category, N is quasi-isomorphic to the complex I^\bullet.
$N \xrightarrow{\sim} I^\bullet = (0 \to I^0 \to I^1 \to \cdots)$
By definition, R\mathrm{Hom}(M, N) = \mathrm{Hom}(M, I^\bullet), the complex with terms \mathrm{Hom}_A(M, I^n).
$R\mathrm{Hom}(M, N) = \mathrm{Hom}_A(M, I^\bullet)$
The cohomology of this complex in degree n is by definition \mathrm{Ext}^n_A(M, N).
$H^n(R\mathrm{Hom}(M, N)) = H^n(\mathrm{Hom}(M, I^\bullet)) = \mathrm{Ext}^n_A(M, N)$

Answer: Ext^n_A(M,N) is recovered as the n-th cohomology of the derived hom RHom(M,N), computed via an injective resolution of N.

Practice Problems

Difficulty 9/10

Prove that a morphism f: X -> Y in D(A) is an isomorphism if and only if the induced maps H^n(f): H^n(X) -> H^n(Y) are isomorphisms for all n \in Z.

Difficulty 10/10

Explain how the cone construction in D^b(A) generalizes the cokernel in A, and give an example.

Difficulty 10/10

Show that for an abelian category A with enough injectives, the canonical functor A -> D(A) is fully faithful.

Common Mistakes

Common Mistake

Thinking D(A) has the same objects as K(A) but with more morphisms — in fact, D(A) is a localization that formally adds inverses to quasi-isomorphisms.

Morphisms in D(A) are equivalence classes of roofs X <- X' -> Y where X' -> X is a quasi-isomorphism. This is not simply adding morphisms but identifying complexes with the same cohomology.

Common Mistake

Confusing the shift X[1] with the suspension: the convention X[1]^n = X^{n+1} (left shift) is standard in algebraic geometry but is opposite to the topology convention in some references.

Always check the convention: in Hartshorne and SGA, X[n]^k = X^{n+k}. In some topology texts, X[1]^k = X^{k-1}.

Historical Background

Derived categories were introduced by Alexander Grothendieck and his student Jean-Louis Verdier in the early 1960s, as part of the framework needed for Grothendieck duality. Verdier's 1967 thesis gave a complete treatment. The subject was later developed by Beilinson, Bernstein, Deligne, and Gabber for perverse sheaves, and by Bondal and Orlov for semiorthogonal decompositions of derived categories of coherent sheaves — now central to mirror symmetry.

1963
Grothendieck and Verdier introduce derived categories
Alexander Grothendieck, Jean-Louis Verdier
1967
Verdier's thesis gives a systematic account of derived categories
Jean-Louis Verdier
1982
BBD (Beilinson-Bernstein-Deligne) develop perverse sheaves via derived categories
Alexander Beilinson, Joseph Bernstein, Pierre Deligne
1990
Bondal-Kapranov introduce DG-enhancements; Bondal-Orlov develop semiorthogonal decompositions
Alexei Bondal, Dmitri Orlov

Summary

D(A) is the localization of the homotopy category K(A) at quasi-isomorphisms, formally inverting chain maps that induce cohomology isomorphisms.
Distinguished triangles X -> Y -> Z -> X[1] replace short exact sequences; the shift functor [1] plays the role of suspension.
Derived functors RF, LF are exact functors between derived categories whose cohomologies recover classical derived functors R^nF, L_nF.
D^b(Coh(X)) for a smooth projective variety X is a rich invariant encoding geometry; semiorthogonal decompositions and exceptional collections are key structural tools.
Grothendieck duality f^! \dashv Rf_* vastly generalizes Serre duality and is formulated naturally in derived categories.

References

BookWeibel, C. An Introduction to Homological Algebra. Cambridge University Press, 1994.
BookHuybrechts, D. Fourier-Mukai Transforms in Algebraic Geometry. Oxford University Press, 2006.
PaperVerdier, J.-L. Des catégories dérivées des catégories abéliennes. Astérisque, 1996.