Efficient generation of zero dimensional ideals in polynomial rings

Abstract

Let A be a commuta t ive noether ian ring and I an ideal in A. It is a wel l -known fact that It(!/I 2) 2 (see [5]). When n = 1 and R is a regular local ring such as a power series r ing over a field or a regular spot over an infinite perfect field, Bha twadekar has shown that It(Ill 2) = # ( I ) for all maximal i d e a l s / i n R [T] (see [2]). In this paper we generalise these results for the case when I is a zero d imens iona l ideal of the fol lowing kind" I. Fo r any zero d imensional ideal I in A=R[T 1 . . . . . T,], we have It(I/I2)=it(I) if n_>2. 2. Let I b e a zero d imensional ideal in A = R[T] . Then It(I/I2)=#(I) in each o f the following cases : (a) R is a power series ring over a field. (b) R is a regular local r ing of an affine a lgebra over an infinite perfect field. In Sect. 2, we state some theorems (without proof) which we require later. In Sect. 3, we prove the analogues of Dav i s -Ge rami t a theorems. In Sect. 4, we prove the result for a power series r ing or for a regular local r ing of an affine algebra. The results of this pape r go th rough for Laurent po lynomia l rings.