= Codimension-two extension of regular functions on a normal variety
{title2=$\Gamma(X,\mathcal O_X)\cong\Gamma(X\setminus Z,\mathcal O_X)$}
For an integral <normal variety> that is <Noetherian>, removing a closed subset of codimension at least two does not create new <regular functions>. Affine-locally, a normal Noetherian domain is the intersection of its localizations at height-one primes inside its <fraction field>. The complement still contains every height-one point, so a <regular function> on it lies in each such <localization> and therefore in the original ring. On projective space this also follows directly by excluding <irreducible polynomial> factor of the denominators on each polynomial <affine chart>.
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