Left exact functor

In homological algebra, an exact functor is a functor that preserves exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.

Let P and Q be abelian categories, and let F: P→Q be a covariant additive functor (so that, in particular, F(0)=0). Let

be a short exact sequence of objects in P.

We say that F is

If G is a contravariant additive functor from P to Q, we can make a similar set of definitions. We say that G is

It is not always necessary to start with an entire short exact sequence 0→A→B→C→0 to have some exactness preserved; it is only necessary that part of the sequence is exact. The following statements are equivalent to the definitions above:

Note, that this does not work for half-exactness. The corresponding condition already implies exactness, since you can apply it to exact sequences of the form 0→A→B→C and A→B→C→0. Thus we get:

Every equivalence or duality of abelian categories is exact.

The most basic examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then F_A(X) = Hom_A(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups. The functor F_A is exact if and only if A is projective. The functor G_A(X) = Hom_A(X,A) is a contravariant left-exact functor; it is exact if and only if A is injective.

If k is a field and V is a vector space over k, we write V* = Hom_k(V,k). This yields a contravariant exact functor from the category of k-vector spaces to itself. (Exactness follows from the above: k is an injective k-module. Alternatively, one can argue that every short exact sequence of k-vector spaces splits, and any additive functor turns split sequences into split sequences.)

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