The following is a quotation from Bosch「Algebraic Geometry and Commutative Algebra」(6.6 Theorem 5)

Let $A$ be a ring, $X = Spec \ A$ its spectrum, and $M$ an $A$-module. Then the functor $$ \textbf{D}(X) \to \textbf{A-Mod}, \quad D(f) \to M_f = M \otimes_A A_f $$ mapping an inclusion of basic open subsets $D(f) \subset D(g)$ to the canonical map $M_g \to M_f$ obtained via localization extends to a sheaf of $\mathcal{O}_X$-modules $\widetilde{M}$ on $X$, called the $\mathcal{O}_X$-module sheaf associated to $M$.

My question is : I want to know why the author did not define it by $$(*) \quad \widetilde{M}(U) = M \otimes_A \mathcal{O}_X(U) .$$ I think this definition $(*)$ works well.

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    $\begingroup$ Over a principal open set $D(f)$, regarding $A$ as a module over itself yields $\mathcal{O}_X(D(f)) = A_f$, so this definition includes the structure sheaf as a special case. Over an arbitrary open set things are harder to describe, so one typically defines it as a limit over principal open sets containing it. $\endgroup$ – DKS Apr 16 at 22:51

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