 Global regular functions on a positive-dimensional projective complete intersection

For a length-$r$ homogeneous regular sequence of positive degrees in ${\mathbb{P}}_k^N$, 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.