A note on the serre dimension of polynomial rings

Abstract

A commutative noetherian ring R is said to have Serre dimension __t+ 1 has a unimodular element, i.e. P = R p ~ ) P ' , for some p e P. Serre's classically well-known theorem asserts that the S e r r e d i m R < d i m R . As pointed out by Plumstead in [P], the Eisenbud-Evans theorem shows that S e r r e d i m R _ G e n e r a l i s e d d i m R . This enabled him to prove that the Serre dim R [X] ___ dim R. Recently, S.M. Bhatwadekar and Amit Roy (see [1]) extended Plumstead's arguments to show that Serre dim R [ X l, . . . , Xn] < d i m R, thereby settling an old conjecture of Bass. In this note we present a different view point of the analysis 'in the fourth corner ' , based on a combination of a theorem of Swan and a variant of a theorem of Roitman. The idea in [1] is to make the following proposition of Plumstead available in the present context;