Exact functor
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In homological algebra, an exact functor is one which preserves exact sequences. Formally, let C and D be abelian categories, and let F:C→D be a functor. Let 0→A→B→C→0 be a short exact sequence. We say that F is
- half-exact if F(A)→F(B)→F(C) is exact.
- 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 functor from C to D, 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.
In fact, it is not always necessary to start with a short exact sequence 0→A→B→C→0 to have some exactness preserved. It is equivalent to say
- 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.
- F is exact if A→B→C exact implies F(A)→F(B)→F(C) 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.
- G is exact if A→B→C exact implies G(C)→G(B)→G(A) exact.
Examples
The most important examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups. The functor FA is exact if and only if A is projective. The functor GA(X) = HomA(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* = Homk(V,k). This yields an 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 HT from the abelian category of all left R-modules to Ab by using the tensor product over R: HT(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 BA consisting of all functors from A to B. If A is a given object of A, then we get a functor EA from BA to B by evaluating functors at A. This functor EA is exact.
Some facts
Every equivalence or duality of abelian categories is exact.
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.
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.