Existence of unimodular elements and cancellation of projective modules over Rees-like algebras and its various extensions

Cancellation of projective modules over regular rings with comparability

On the existence of unimodular elements and cancellation of projective modules over noetherian and non-noetherian rings

Generating modules efficiently: Theorems from algebraic K-theory

A note on cancellation of projective modules

We study the cancellation property of projective modules of rank $2$ with a\ntrivial determinant over Noetherian rings of dimension $\\leq 4$. If $R$ is a\nsmooth affine algebra of dimension $4$ over an algebraically closed field $k$\nsuch that $6 \\in k^{\\times}$, then we prove that stably free $R$-modules of\nrank $2$ are free if and only if a Hermitian $K$-theory group $\\tilde{V}_{SL}\n(R)$ is trivial.\n

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

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).

On projective modules over polynomial rings

Projective modules over overrings of polynomial rings

The Euler class groups of polynomial rings and unimodular elements in projective modules

Unimodular elements in projective modules

A note on projective modules over real affine algebras

(1) Let $R$ be a commutative Noetherian ring of dimension $n$ and $P$ a projective $R[X_1,\ldots ,X_m]$-module of rank $n$. In this paper, we associate an obstruction for $P$ to split off a free summand of rank one. (2) Let $R$ be a local ring and $R[X]\subset A\subset R[X,X^{-1}]$. Let $P$ and $Q$ be two projective $A$-modules with $\text {rank}(Q)\lt \text {rank}(P)$. If $Q_f$ is a direct summand of $P_f$ for some special monic polynomial $f\in R[X]$, then $Q$ is also a direct summand of $P$.

A Cancellation Theorem for Projective Modules over Finitely Generated Rings

On a question of Nori: Obstructions, improvements, and applications