This is a statement made in Mumford, Oda Algebraic Geometry II.(Chpt 1, Sec 2. Prop 2.7) http://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf

If $M_a$ is any collection of $R$ modules and denote $X=Spec(R)$, then $\overline{\oplus_aM_a}=\oplus_a\overline{M_a}$ where $\overline{M}$ denotes formation of sheaf of $O_X$ modules.

The proof goes along by using $Spec(R_f)$ affine which enforces quasi-compact. Then both sides of equation are canonically identified.(It is not hard to check this equality.) Then extension to a unique isomorphic sheaf by $\mathcal{B}$-sheaf extension.

$\textbf{Q:}$ Isn't this is sort of saying infinite direct sum of quasi-coherent sheaves of modules is quasi-coherent over affine scheme? Or am I misunderstanding the statement.

$\textbf{Q':}$ What is the example of infinite direct sum of sheaves of free module over affine scheme fails to be a sheaf?


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