A theorem of Eakin and Sathaye and Green’s hyperplane restriction theorem

The purpose of these notes is to show how the following Theorem 2.1, due to Eakin and Sathaye, can be viewed, after some standard reductions, as a corollary of Green’s hyperplane restriction theorem.

Theorem 2.1 (Eakin-Sathaye) Let (R,m) be a quasi-local ring with infinite residue field. Let I be an ideal of R. Let n and r be positive integers. If the

number of minimal generators of Ii, denoted by v(Ii), satisfies

v(Ii) < (

i+ r r

) ,

then there are elements h1, . . . ,hr in I such that Ii = (h1, . . . ,hr)Ii−1.