Stability theorems for overrings of polynomial rings

Abstract

Let R be a polynomial ring over a field. Projective modules over rings of the type R IX, Y] / ( r -X Y) where r is a non-zero element of R have been considered by Murthy, Swan, Weibel (see [Mu] and [W]). These rings lie between R[X] and R[ X ,X 1] and even when they are regular, projective modules over them need not be free ([Mu], Example 6.2). This leads us to study stability properties of projective modules over rings A which lie between R [X] and R[X,X-1] when R is any commutative noetherian ring. We prove stability theorems for projective modules over such rings A in w These results (Theorems 4.2 and 4.3) have been proved for A=R[X] by Plumstead [P] and for A = R [ X , X -1] by Mandal [Ma]. In w we prove similar results for the allied class of rings D of the type R[X, Y]/(XY). Stability theorems for GL,(A) and GL,(D) are proved in w When A=R[X] or R[X,X-1], our Theorems 6.2(i) and 6.4(i) had already been proved by Suslin [-S]. Finally in w 7 we study Pic(R[X,Y]/(r-XY)) when R is a PID. Theorem 7.1 extends a result of Murthy ([Mu], Corollary 5.3). It would be interesting to know whether Theorems 4.1 and 4.2 can be extended to the polynomial ring A[T 1 ..... Tm] (where A lies between R[X] and R[X,X-1]) . More precisely, for a projective A[T 1 .... ,T,,]-module P of rank > dim A we would like to know if the following statements are true: (i) P has a unimodular element. (ii) P has the cancellation property. We have been able to show that when A = R [ X ] , the statement (i) is true ([B-R], Theorem 3.1).