= Global regular functions on a positive-dimensional projective complete intersection
{title2=$H^0(X,\mathcal O_X)=k$}
For a length-$r$ homogeneous regular sequence of positive degrees in $\mathbb P^N_k$, with $r<N$, the <global regular functions> on its <projective complete intersection> are exactly $k$. In the successive restriction sequences, <intermediate cohomology vanishing for a projective complete intersection> makes the first-cohomology group on the preceding stage zero. Inductively all negative twists have zero global sections, so at twist zero the restriction of constants is an isomorphism. The positive-dimension assumption is essential: a zero-dimensional complete intersection can have more global functions, including nilpotents.
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