Exact functor

From Academic Kids

In homological algebra, an exact functor is one which preserves exact sequences. Formally, let C and D be abelian categories, and let F:CD be a functor. Let 0ABC0 be a short exact sequence. We say that F is

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

In fact, it is not always necessary to start with a short exact sequence 0ABC0 to have some exactness preserved. It is equivalent to say

  • F is left-exact if 0ABC exact implies 0F(A)F(B)F(C) exact.
  • F is right-exact if ABC0 exact implies F(A)F(B)F(C)0 exact.
  • F is exact if ABC exact implies F(A)F(B)F(C) exact.
  • G is left-exact if ABC0 exact implies 0G(C)G(B)G(A) exact.
  • G is right-exact if 0ABC exact implies G(C)G(B)G(A)0 exact.
  • G is exact if ABC 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) = TX. 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.

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