I have an exact functor $F:\mathcal{A}\to \mathcal{B}$ between abelian categories. Does it necessary that $F$ have both left adjoint and right adjoint? if so why? I am new to abelian categories I am not able to justify/think on it.
2026-03-26 14:17:20.1774534640
Why exact functor have both adjoints?
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This is not true in general. If $P$ is projective finitely generated $R$-module, then the functor
$$Hom(P,-): Mod_R \to Ab$$
is exact, however, it only preserves direct sums if $P$ is finitely generated. So if $P$ is not finitely generated, $Hom(P,-): Mod_R \to Ab$ is exact, but does not admit a left adjoint.