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Let $A$ be a commutative ring with unit, $X=\operatorname{Spec}A$, $\mathcal{B}$ the base of open subsets on $X$ made up of the principal open subsets. Then Liu's book "The Algebraic Geometry and Arithmetic Curves" gives a proof of the following (page 42):

$\mathcal{O}_X$ is a $\mathcal{B}$-sheaf of rings. It therefore induces a sheaf of rings $\mathcal{O}_X$ on $\operatorname{Spec}A$, and we have $\mathcal{O}_X(X)=A$.

In the proof Liu verifies the uniqueness and glueing local sections for the open set $U=X$.

Is this special case needed in proving the general case where $U$ is an arbitrary principal open set or do we need completely different proof in the general case?

studying
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  • Hint: Basic-open subsets are affine, and localizations commute with localizations. 2nd Hint: Forget about the "$\mathcal{B}$-sheaf" business, since it's quite complicated when compared to the direct definition of the structure sheaf (see e.g. Hartshorne's book). I'll probably never get why this approach is so popular. In the end, it doesn't even work properly (http://math.stackexchange.com/questions/390347/). – Martin Brandenburg Dec 23 '13 at 21:16
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    @MartinBrandenburg I thought the $\mathcal B$-sheaves were a useful tool for gluing sheaves together. Furthermore, isn't checking that $\mathcal O_X(U)=S_U^{-1} A$ for principal $U$ amount to the same thing as checking that $\mathcal O_X(U)=S_U^{-1} A$ is a $\mathcal B$-sheaf, so we might as well use the general result on sheaves generated by $\mathcal B$-sheaves? – Vladimir Sotirov Dec 23 '13 at 22:39
  • Gluing sheaves is important, but gluing them on bases is not so important and quite ugly. – Martin Brandenburg Dec 24 '13 at 00:52

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