= Localization of global sections on a principal open
{title2=$\Gamma(X,\mathcal O_X)_f\cong\Gamma(X_f,\mathcal O_X)$}
For a quasi-compact <separated scheme> and a global <regular function> $f$, <global sections> on $X_f$ are the <localization> of <global sections> by $f$. Choose a finite affine <open cover>. Its intersections are affine, so the <sheaf gluing axiom> describes <global sections> as the <kernel> of the difference map between two finite products of affine <coordinate rings>. <Exactness of localization> preserves this <kernel>, and <localization> commutes with those finite products. Localizing the same cover gives the claimed <isomorphism>. This applies before knowing whether $X$ itself is affine.
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