In the Lemma 5.1.3 of Liu Qing's book on algebraic geometry, he uses $O_{X}^{(I)}$ which the direct sum of $O_{X}$indexed by $I$.
What's the global section of this sheaf? Is it $\bigoplus O_{X}(X)$? But I think this is impossible since when $I$ is infinite we can suppose $X=\bigsqcup U_{i}$ and glue $1_{U_{i}}$ together to get a global section which can't be an element in the direct sum.
Or we just get a presheaf from $O_{X}^{(I)}$ and we consider its associate sheaf instead? If so, is there a big difference between direct sum of infinite sheaves and product of infinite sheaves?