# Why Bosch define $O_X$-module sheaf associated to M on open basis?

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.

• 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. – DKS Apr 16 at 22:51