Exactness, and left, right, and exact functors

(exact sequences and exact functors)

In an abelian category $\mca$, a sequence $A \mapsvia{f} B \mapsvia{g} C$ is exact at $B$ if $\ker g = \im f$. A sequence \[ 0 \to A \mapsvia{f} B \mapsvia{g} C \to 0 \] is a short exact sequence if $f$ is injective, $g$ is surjective, and $\ker g = \im f$.

An additive functor $F: \mca \to \mcb$ is left exact if for every short exact sequence $0 \to A \to B \to C \to 0$ the sequence \[ 0 \to F(A) \to F(B) \to F(C) \] is exact, and right exact if \[ F(A) \to F(B) \to F(C) \to 0 \] is exact. $F$ is exact if it is both, in which case short exact sequences go to short exact sequences.

The functor $\Hom_A(N, \wait)$ is left exact and $\wait \tensor_A N$ is right exact, since $\wait\tensor_AN$ is left adjoint to $\Hom_A(N,\wait)$ (D-DEFADJ). Over $A = \ZZ$ with $N = \ZZ/2$, apply both to $0 \to \ZZ \mapsvia{2} \ZZ \to \ZZ/2 \to 0$. The map $\Hom(\ZZ/2,\ZZ)=0\to\Hom(\ZZ/2,\ZZ/2)=\ZZ/2$ is not surjective, and the map $\ZZ\tensor\ZZ/2\to\ZZ\tensor\ZZ/2$ induced by multiplication by $2$ is the zero map $\ZZ/2\to\ZZ/2$, which is not injective. The long exact sequences of $\Ext^i_\ZZ(\ZZ/2,\wait)$ and $\Tor_i^\ZZ(\wait,\ZZ/2)$ continue these sequences, with $\Ext^1_\ZZ(\ZZ/2,\ZZ)\cong\ZZ/2$ and $\Tor_1^\ZZ(\ZZ/2,\ZZ/2)\cong\ZZ/2$.

What does it mean for a sequence to be exact at an object, and what is a short exact sequence?

Define left exact, right exact, and exact for an additive functor.

Give a left exact functor that is not exact, and a right exact one that is not exact.

Dependencies

D-DEFABCATAbelian categoriesuses

Backlinks

D-DEFDERIVDerived functorsuses
D-DEFFLATFlat modulesuses
D-DEFINJOBInjective objects, injective resolutions, and enough injectivesuses
T-DEFRAPLRAPL and LAPC, and the exactness of $\Hom$ and $\tensor$uses
T-DEFFHHFThe FHHF theorem, on functors and cohomologyuses