= Affineness from a unit-ideal principal affine cover
{title2=$\sum_i g_if_i=1,\quad X_{f_i}\text{ affine}\ \Longrightarrow\ X\text{ affine}$}
A classical <variety> is affine if finitely many global <regular functions> generate the <unit ideal> and their nonvanishing opens are affine. <Localization of global sections on a principal open> identifies their <coordinate rings> with $A_{f_i}$, where $A=\Gamma(X,\mathcal O_X)$. Choose a finite-type subalgebra of $A$ containing the functions, their unit-ideal coefficients and numerators of generators of every $A_{f_i}$. Its affine variety has the same principal affine charts, whose <isomorphisms> glue globally. This avoids assuming finite generation of $A$ before affineness is known.
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