S.M. Bhatwadekar

The Euler Class Group of a Noetherian Ring

For a commutative Noetherian ring A of finite Krull dimension containing the field of rational numbers, an Abelian group called the Euler class group is defined. An element of this group is attached to a projective A-module of rank = dimA and it is shown that the vanishing of this element is necessary and sufficient for P to split off a free summand of rank 1. As one of the applications of this result, it is shown that for any n-dimensional real affine domain, a projective module of rank n (with trivial determinant), all of whose generic sections have n generated vanishing ideals, necessarily splits off a free direct summand of rank 1.

On residual variables and stably polynomial algebras

Let R be a commutative noetherian ring and let R[X,Y] be a polynomial algebra in two variables over R. Let W be an element of R[X,Y]. In this paper we show that R[X,Y] is a stably polynomial algebra over R[W] if and only if W is a residual variable in R[X,Y]. Moreover, in this case, if either R contains the field of rationals or if Rred is seminormal, then W is a variable in R[X,Y].

A cancellation theorem for projective modules over affine algebras over C 1 -fields

Let k be a C1-field of characteristic zero. Let A be an affine algebra of dimension d⩾2 over k. In this set up, Suslin proved that the free module Ad is cancellative (in other words, stably free A-modules of rank d are free). In this note we show that, in fact, all finitely generated projective A-modules of rank d are cancellative.

On A∗-fibrations

Abstract In this paper we investigate minimal sufficient fibre conditions for a finitely generated flat algebra over a noetherian integral domain to be locally A ∗ or at least an A ∗ -fibration. We also describe the structure of finitely generated locally A ∗ -algebras.

Structure of A2-fibrations over one-dimensional noetherian domains

Let R be a one-dimensional noetherian domain containing the field Q of rational numbers. Let A be an A2-fibration over R. Then there exists H ϵ A such that A is an A1-fibration over R[H]. As a consequence, if ΩAR is free then A = R[2].

A note on the cancellation property of k[X, Y]

In [On affine-ruled rational surfaces, Math. Ann.255(3) (1981) 287–302], Russell had proved that when k is a perfect field of positive characteristic, the polynomial ring k[X, Y] is cancellative. In this note, we shall show that this cancellation property holds even without the hypothesis that k is perfect.

On Automorphisms of Modules over Polynomial Rings

Write f = Xfi - bXf . Then f is in XP* and f(q) = 0. Clearly, 1 + qf is in EL( P, X) and it is a lift of 1 +pg. In a similar way we can lift 1 +pg when g is unimodular. This completes the proof of (2.8). The following is a variant of a proposition of Lindel ([L, 2.71; see 4.3 for the statement). Like the proposition of Lindel in his paper, this proposi- tion plays a key roll in the proof of our main theorems ((3.1) and (3.2)). We shall go into detailed discussions on Lindel’s proposition [L, 2.71 in our later sections (Sects. 4, 5, 6). And now we shall state our proposition. (2.9)

On algebras which are locally

Let R be a Noetherian normal domain. Call an R-algebra A “locally A1 in codimension-one” if RP ⊗R A is a polynomial ring in one variable over RP for every height-one prime ideal P in R. We shall describe a general structure for any faithfully flat R-algebra A which is locally A1 in codimensionone and deduce results giving sufficient conditions for such an R-algebra to be a locally polynomial algebra. We also give a recipe for constructing R-algebras which are locally A1 in codimension-one. When R is a normal affine spot (i.e., a normal local domain obtained by a localisation of an affine domain), we give criteria for a faithfully flat R-algebra A, which is locally A1 in codimensionone, to be Krull and a further condition for A to be Noetherian. The results are used to construct intricate examples of faithfully flat R-algebras locally A1 in codimension-one which are Noetherian normal but not finitely generated.