Example additive functor which is neither right exact nor left exact.

Question

I have a problem a bout finding an example for this problem

Give an example of an additive functor $T : Ab\rightarrow Ab$ which is neither right exact nor left exact.

I can not think in one example for that, i would appreciate any hint about this.

Answer 1

Just so that this gets answered, let's apply my suggestion from the comments. Though it's even easier if we take $T$ to be the direct sum of a left exact and a right exact functor, as follows.

Define $T$ by $T(A) = \newcommand\Ab{\mathbf{Ab}}\newcommand\Z{\Bbb{Z}} \Ab(\Z/2,A) \oplus (A\otimes \Z/2)$.

Then consider $$0\to \Z \newcommand\toby\xrightarrow\toby{2} \Z \to \Z/2\to 0. $$ tensoring with $\Z/2$ gives $$0\to \Z/2 \toby{0} \Z/2\toby{1} \Z/2 \to 0,$$ which is no longer left exact. Taking $\Ab(\Z/2,-)$ we instead get $$0\to 0 \to 0 \to \Z/2 \to 0,$$ which is no longer right exact.

Taking the direct sum, we get $$0\to \Z/2 \toby{0}\Z/2 \toby{\iota_1} (\Z/2)^2 \to 0,$$ which is not left or right exact.