Let A = Mod(R) be an abelian category. A complex in A is a sequence of objects and morphisms in A
… → Mi-1 →di-1 Mi →di → Mi+1 → …
such that di ◦ di-1 = 0 ∀ i. We denote such a complex by M.
A morphism of complexes f : M → N is a sequence of morphisms fi : Mi → Ni in A, making the following diagram commute, where diM, diN denote the respective differentials:
We let C(A) denote the category whose objects are complexes in A and whose morphisms are morphisms of complexes.
Given a complex M of objects of A, the ith cohomology object is the quotient
Hi(M) = ker(di)/im(di−1)
This operation of taking cohomology at the ith place defines a functor
Hi(−) : C(A) → A,
since a morphism of complexes induces corresponding morphisms on cohomology objects.
Put another way, an object of C(A) is a Z-graded object
M = ⊕i Mi
of A, equipped with a differential, in other words an endomorphism d: M → M satisfying d2 = 0. The occurrence of differential graded objects in physics is well-known. In mathematics they are also extremely common. In topology one associates to a space X a complex of free abelian groups whose cohomology objects are the cohomology groups of X. In algebra it is often convenient to replace a module over a ring by resolutions of various kinds.
A topological space X may have many triangulations and these lead to different chain complexes. Associating to X a unique equivalence class of complexes, resolutions of a fixed module of a given type will not usually be unique and one would like to consider all these resolutions on an equal footing.
A morphism of complexes f: M → N is a quasi-isomorphism if the induced morphisms on cohomology
Hi(f): Hi(M) → Hi(N) are isomorphisms ∀ i.
Two complexes M and N are said to be quasi-isomorphic if they are related by a chain of quasi-isomorphisms. In fact, it is sufficient to consider chains of length one, so that two complexes M and N are quasi-isomorphic iff there are quasi-isomorphisms
M ← P → N
For example, the chain complex of a topological space is well-defined up to quasi-isomorphism because any two triangulations have a common resolution. Similarly, all possible resolutions of a given module are quasi-isomorphic. Indeed, if
0 → S →f M0 →d0 M1 →d1 M2 → …
is a resolution of a module S, then by definition the morphism of complexes
is a quasi-isomorphism.
The objects of the derived category D(A) of our abelian category A will just be complexes of objects of A, but morphisms will be such that quasi-isomorphic complexes become isomorphic in D(A). In fact we can formally invert the quasi-isomorphisms in C(A) as follows:
There is a category D(A) and a functor Q: C(A) → D(A)
with the following two properties:
(a) Q inverts quasi-isomorphisms: if s: a → b is a quasi-isomorphism, then Q(s): Q(a) → Q(b) is an isomorphism.
(b) Q is universal with this property: if Q′ : C(A) → D′ is another functor which inverts quasi-isomorphisms, then there is a functor F : D(A) → D′ and an isomorphism of functors Q′ ≅ F ◦ Q.
First, consider the category C(A) as an oriented graph Γ, with the objects lying at the vertices and the morphisms being directed edges. Let Γ∗ be the graph obtained from Γ by adding in one extra edge s−1: b → a for each quasi-isomorphism s: a → b. Thus a finite path in Γ∗ is a sequence of the form f1 · f2 ·· · ·· fr−1 · fr where each fi is either a morphism of C(A), or is of the form s−1 for some quasi-isomorphism s of C(A). There is a unique minimal equivalence relation ∼ on the set of finite paths in Γ∗ generated by the following relations:
(a) s · s−1 ∼ idb and s−1 · s ∼ ida for each quasi-isomorphism s: a → b in C(A).
(b) g · f ∼ g ◦ f for composable morphisms f: a → b and g: b → c of C(A).
Define D(A) to be the category whose objects are the vertices of Γ∗ (these are the same as the objects of C(A)) and whose morphisms are given by equivalence classes of finite paths in Γ∗. Define a functor Q: C(A) → D(A) by using the identity morphism on objects, and by sending a morphism f of C(A) to the length one path in Γ∗ defined by f. The resulting functor Q satisfies the conditions of the above lemma.
The second property ensures that the category D(A) of the Lemma is unique up to equivalence of categories. We define the derived category of A to be any of these equivalent categories. The functor Q: C(A) → D(A) is called the localisation functor. Observe that there is a fully faithful functor
J: A → C(A)
which sends an object M to the trivial complex with M in the zeroth position, and a morphism F: M → N to the morphism of complexes
Composing with Q we obtain a functor A → D(A) which we denote by J. This functor J is fully faithful, and so defines an embedding A → D(A). By definition the functor Hi(−): C(A) → A inverts quasi-isomorphisms and so descends to a functor
Hi(−): D(A) → A
establishing that composite functor H0(−) ◦ J is isomorphic to the identity functor on A.