Table of contents |

2 Construction and first properties 3 Variations 4 Applications 5 Naturality 6 Generalization |

It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations.

Suppose we are given a covariant left exact functor *F* : **A** → **B** between two abelian categories **A** and **B**. If

The crucial assumption we need to make about our abelian category **A** is that it have *enough injectives*, meaning that for every object *A* in **A** there exists a monomorphism *A* → *I* where *I* is an injective object in **A**.

The right derived functors of the covariant left-exact functor *F* : **A** → **B** is then defined as follows. Start with an object *X* of **A**. Because there are enouch injectives, we can construct a long exact sequence of the form

The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of the snake lemma.

If *X* is itself injective, then we can choose the injective resolution 0 → *X* → *X* → 0, and we obtain that *R ^{i}F*(

- .

**Sheaf cohomology.** If *X* is a topological space, then the category of all sheaves of abelian groups on *X* is an abelian category with enough injectives (a result of Grothendieck). The functor which assigns to each such sheave *L* the group *L*(*X*) of global sections is left exact, and the right derived functors are the sheaf cohomology functors, usually written as *H*^{ i}(*X*,*L*). Slightly more generally: if (*X*, O_{X}) is a ringed space, then the category of all sheaves of O_{X}-modules is an abelian category with enough injectives, and we can construct sheaf cohomology as the right derived functors of the global section functor.

**Ext functors.** If *R* is a ring, then the category of all left *R*-modules is an abelian category with enough injectives. If *A* is a fixed left *R*-module, then the functor Hom(*A*,-) is left exact, and its right derived functors are the Ext functors Ext^{i}(*A*,*B*).

**Tor functors.** The category of left *R*-modules also has enough projectives. If *A* is a fixed left *R*-module, then the tensor product with *A* gives a right exact covariant functor; its left derivatives are the Tor functors Tor_{i}(*A*,*B*).

**Group cohomology.** Let *G* be a group. A *G*-module *M* is an abelian group *M* together with a group action of *G* on *M* as a group of automorphisms. This is the same as a module over the group ring **Z***G*. The *G*-modules form an abelian category with enough injectives. We write *M*^{G} for the subgroup of *M* consisting of all elements of *M* that are held fixed by *G*. This is a left-exact functor, and its right derived functors are the group cohomology functors, typically written as H^{ i}(*G*,*M*).

Derived functors and the long exact sequences are "natural" in several technical senses.

First, given a commutative diagram of the form

Second, suppose η : *F* → *G* is a natural transformation from the left exact functor *F* to the left exact functor *G*. Then natural transformations *R ^{i}*η :

is induced.