= Affine scheme reconstruction from global sections
{title2=$X\cong\operatorname{Spec}\Gamma(X,\mathcal O_X)$}
For an <affine scheme> $X=\operatorname{Spec}A$, its ring of <global sections> is $A$. A <morphism of schemes> between affine schemes is determined contravariantly by its map on these rings. Consequently a morphism between affine schemes is an <isomorphism of schemes> exactly when that ring map is an isomorphism. The affine hypothesis is essential, as the <punctured affine plane> shows.
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