In mathematics, particularly homological algebra, an exact functor is a functor that preserves short 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.
Further, we say that F is If G is a contravariant additive functor from P to Q, we similarly define G to be It is not always necessary to start with an entire short exact sequence 0→A→B→C→0 to have some exactness preserved.
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 FA(X) = HomA(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups.
[1] The functor FA is exact if and only if A is projective.
This yields a contravariant exact functor from the category of k-vector spaces to itself.
Alternatively, one can argue that every short exact sequence of k-vector spaces splits, and any additive functor turns split sequences into split sequences.)
The covariant functor that 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.
The functor HT is exact if and only if T is flat.
Proof: It suffices to show that if i is an injective map of
Therefore, it suffices to show that if a pure tensor
In general, if T is not flat, then tensor product is not left exact.
For example, consider the short exact sequence of
While tensoring may not be left exact, it can be shown that tensoring is a right exact functor: Theorem: Let A,B,C and P be R-modules for a commutative ring R having multiplicative identity.
be a short exact sequence of R-modules.
Then is also a short exact sequence of R-modules.
, where f is the inclusion and g is the projection, is an exact sequence of R-modules.
is also a short exact sequence of R-modules.
given by R-linearly extending the map defined on pure tensors:
So, the kernel of this map cannot contain any nonzero pure tensors.
and n is the highest power of 2 dividing m. We prove a special case: m=12.
, A,B,C,P are R=Z modules by the usual multiplication action and satisfy the conditions of the main theorem.
By the exactness implied by the theorem and by the above note we obtain that
The last congruence follows by a similar argument to one in the proof of the corollary showing that
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 iff it turns finite colimits into limits; a contravariant functor is right exact iff 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.
In SGA4, tome I, section 1, the notion of left (right) exact functors are defined for general categories, and not just abelian ones.
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. The exact functors between Quillen's exact categories generalize the exact functors between abelian categories discussed here.