# Dual of locally free sheaves commute with direct sums

algebraic-geometrycoherent-sheaves

For any locally free sheaf $$\mathcal F$$ on a scheme $$(X,\mathcal O_X)$$ of finite rank, its dual is defined as $$\mathcal F^{\vee}:=\mathscr Hom(\mathcal F,\mathcal O_X)$$. So,

if $$\mathcal F,\mathcal G$$ are locally free sheaves, what is the dual of $$\mathcal F\oplus\mathcal G$$?

I would like to think that $$(\mathcal F\oplus\mathcal G)^{\vee}=\mathcal F^{\vee}\oplus\mathcal G^{\vee}$$, since for every short exact sequence of locally free
sheaves on $$X$$ the $$\mathscr Hom(-,\mathcal O_X)$$ functor is exact, but I think that this is not true because short exact sequences of locally free sheaves don't necessarily split, (see this answer: Short exact sequence of vector bundles vs locally free sheaves).

In the particular if $$X$$ is a projective variety, what is the dual of $$\mathcal O(1)\oplus\mathcal O(2)$$ ?

Any additive functor between abelian categories sends split exact sequences to split exact sequences. For example you can see a proof here for abelian groups. Since $$^{\vee}$$ is an additive functor, it sends the split sequence
$$0 \to \mathcal F \to \mathcal F \oplus \mathcal G \to \mathcal G \to 0$$
$$0 \to \mathcal G^{\vee} \to \mathcal (F \oplus \mathcal G)^{\vee} \to \mathcal F^{\vee} \to 0$$.
and since this is again split, $$(F \oplus \mathcal G)^{\vee} \cong F^\vee \oplus \mathcal G^{\vee}$$. Note that this applieas to $$\mathcal O_X$$-modules in general, not only locally free sheaves, and to any other functor of $$\mathcal O_X$$-modules which is additive (almost all of them are)