Half exact functor which is neither right exact nor left exact

Question

A half exact functor is a functor F (between abelian categories) such that for every short exact sequence: $$ 0 \to A \to B \to C \to 0$$ then $$F(A) \to F(B) \to F(C)$$ is exact. Does anyone has an example of a half exact funtor that is not right nor left exact? What about a functor between abelian categories that is not right nor left nor half exact?

Answer 1

A common source of half-exact functors which are neither right-exact nor left-exact is derived functors, which are always half exact by the long exact sequences relating them but are rarely right-exact or left-exact. For instance, if $n>0$ and $A$ is any object of projective dimension $>n$, then the functor $\operatorname{Ext}^n(A,-)$ is half-exact but neither right-exact nor left-exact.

Here's a particularly simple example. Let $\mathcal{C}$ be the category of chain complexes of vector spaces (over your favorite field $k$), let $\mathcal{D}$ be the category of graded vector spaces, and let $F:\mathcal{C}\to\mathcal{D}$ take a chain complex to its homology. Then $F$ is half-exact by the long exact sequence in homology associated to a short exact sequence of chain complexes. But $F$ is neither right-exact nor left exact, precisely because the connecting homomorphisms in those long exact sequences can be nontrivial. Explicitly, if $A$ is the chain complex $0\to 0\to k \to 0$, $B$ is the chain complex $0\to k\stackrel{1}\to k\to 0$, and $C$ is the chain complex $0\to k\to 0\to 0$, then the obvious short exact sequence $0\to A\to B\to C\to 0$ is taken by $F$ to a sequence which is exact only in the middle.

For a simple example of an additive functor which is not half-exact, consider the functor $F:Ab\to Ab$ which sends an abelian group to its subgroup of elements divisible by $2$. To see that it is not half-exact, consider what it does to the short exact sequence $0\to 2\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}/2\to 0$.