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.

Definitions

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

0→A→B→C→0

be a short exact sequence of objects in P.

We say that F is

• half-exact if F(A)→F(B)→F(C) is exact. This is similar to the notion of a topological half-exact functor.
• left-exact if 0→F(A)→F(B)→F(C) is exact.
• right-exact if F(A)→F(B)→F(C)→0 is exact.
• exact if 0→F(A)→F(B)→F(C)→0 is exact.

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

• half-exact if G(C)→G(B)→G(A) is exact.
• left-exact if 0→G(C)→G(B)→G(A) is exact.
• right-exact if G(C)→G(B)→G(A)→0 is exact.
• exact if 0→G(C)→G(B)→G(A)→0 is exact.

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:

• F is left-exact if 0→A→B→C exact implies 0→F(A)→F(B)→F(C) exact.
• F is right-exact if A→B→C→0 exact implies F(A)→F(B)→F(C)→0 exact.
• G is left-exact if A→B→C→0 exact implies 0→G(C)→G(B)→G(A) exact.
• G is right-exact if 0→A→B→C exact implies G(C)→G(B)→G(A)→0 exact.

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:

• F is exact if and only if A→B→C exact implies F(A)→F(B)→F(C) exact.
• G is exact if and only if A→B→C exact implies G(C)→G(B)→G(A) exact.

Examples

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.^[1] The functor F_A is exact if and only if A is projective.^[2] The functor G_A(X) = Hom_A(X,A) is a contravariant left-exact functor;^[3] it is exact if and only if A is injective.^[4]

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.)

If X is a topological space, we can consider the abelian category of all sheaves of abelian groups on X. The functor which associates to each sheaf F the group of global sections F(X) is left-exact.

If R is a ring and T is a right R-module, we can define a functor H_T from the abelian category of all left R-modules to Ab by using the tensor product over R: H_T(X) = T ⊗ X. This is a covariant right exact functor; it is exact if and only if T is flat.

If A and B are two abelian categories, we can consider the functor category B^A consisting of all functors from A to B. If A is a given object of A, then we get a functor E_A from B^A to B by evaluating functors at A. This functor E_A is exact.

Properties and theorems

A covariant (not necessarily additive) functor is left exact if and only if it turns finite limits into limits; a covariant functor is right exact if and only if it turns finite colimits into colimits; a contravariant functor is left exact if and only if it turns finite colimits into limits; a contravariant functor is right exact if and only if it turns finite limits into colimits. A functor is exact if and only if it is both left exact and right exact.

The degree to which a left exact functor fails to be exact can be measured with its right derived functors; the degree to which a right exact functor fails to be exact can be measured with its left derived functors.

Left and right exact functors are ubiquitous mainly because of the following fact: if the functor F is left adjoint to G, then F is right exact and G is left exact.

Generalization

In SGA4, tome I, section 1, the notion of left (right) exact functors are defined for general categories, and not just abelian ones. The definition is as follows:

Let C be a category with finite projective (resp. inductive) limits. Then a functor u from C to another category C′ is left (resp. right) exact if it commutes with finite projective (resp. inductive) limits.

Despite its abstraction, this general definition has useful consequences. For example, in section 1.8, Grothendieck proves that a functor is pro-representable if and only if it is left exact, under some mild conditions on the category C.

Notes

1. Jacobson (2009), p. 98, Theorem 3.1.
2. Jacobson (2009), p. 149, Prop. 3.9.
3. Jacobson (2009), p. 99, Theorem 3.1.
4. Jacobson (2009), p. 156.

References

• Jacobson, Nathan (2009). Basic algebra. 2 (2nd ed.). Dover. ISBN 978-0-486-47187-7.