Formal functions over Grassmannians (Q788788)

The authors prove the following theorem: The field of formal-rational functions \(\hat K\) along a connected closed subscheme of positive dimension in a Grassmannian Grass(n,r) is exactly the field of rational functions on Grass(n,r). The proof uses a reduction to the special case of \({\mathbb{P}}^ 1\) in \({\mathbb{P}}^ n\) (where the result is known) and a lemma which states \(\Phi((X))\cap L(X)=\Phi(X)\) for fields \(\Phi\subset L\) and a finite set of indeterminates X.