Let $\ES{A},\ES{B}$ be abelian categories, $\ES{A}$ having enough injectives, and $F:\ES{A}\to\ES{B}$ be a covariant left-exact functor.

Given an object $A\in \ES{A}$ , we can construct an injective resolution

A completely analogous construction can be carried out for right-exact functors and for contravariant functors exact on either side, but it is traditional to only describe one case, as doing the others mostly consists of reversing arrows (and replacing ``injective'' with projective when appropriate), and the result is that of a left derived functor $L^iF$ for a given right-exact covariant functor $F$ via a projective resolution.

Important properties of the classical derived functors are these: If the sequence $0\to A\to A'\to A''\to 0$ is exact, then there is a long exact sequence

From the definition, one can see immediately that the following are equivalent:

1. $F$ is exact
2. $R^nF(A)=0$ for $n\geq 1$ and all $A\in \ES{A}$ .
3. $R^1F(A)=0$ for all $A\in \ES{A}$ .

However, $R^1F(A)=0$ for a particular $A$ does not imply that $R^nF(A)=0$ for all $n\geq 1$ .

Important examples are:

• The Ext functors $\mathrm{Ext}^n$ are the right derived functors of $\mathrm{Hom}$ .
• The Tor functors $\mathrm{Tor}_n$ are the left derived functors of the tensor product.
• Sheaf cohomology arises as the right derived functors of the global section functor on sheaves.
• Group cohomology arises as the right derived functors of the ``fixed submodule'' functor on the category of $G$ -modules for some group $G$ .
• Étale cohomology arises as the right derived functors of the global sections functor on the category of étale sheaves; this example includes as special cases the previous two.