A note on generators of least degree in Gorenstein ideals

Abstract

Assume R R is a polynomial ring over a field and I I is a homogeneous Gorenstein ideal of codimension g ≥ 3 g\ge 3 and initial degree p ≥ 2 p\ge 2 . We prove that the number of minimal generators ν ( I p ) \nu (I_p) of I I that are of degree p p is bounded above by ν 0 = ( p + g − 1 g − 1 ) − ( p + g − 3 g − 1 ) \nu _0=\binom {p+g-1}{g-1}-\binom {p+g-3}{g-1} , which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension g g and initial degree p p . Further, I I is itself extremal if ν ( I p ) = ν 0 \nu (I_p)=\nu _0 .