Affine Algebra Of Dimension Research Articles

Applications of Swan's Bertini theorem to unimodular rows

Let R be an affine algebra of dimension d≥4 over a perfect field k of char ≠2 and I be an ideal of R. Then(1)MSd+1(R) is uniquely divisible prime to char k if R is reduced and k is infinite with c.d.(k)≤1.(2)Umd+1(R,I)/Ed+1(R,I) has nice group structure if c.d.2(k)≤2.(3)Umd(R,I)/Ed(R,I) has nice group structure if k is algebraically closed of char k≠2,3 and either (i) k=F‾p or (ii) R is normal.

Unimodular rows over affine algebras over algebraic closure of a finite field

In this paper, we prove that if [Formula: see text] is an affine algebra of dimension [Formula: see text] over [Formula: see text] and [Formula: see text] then any unimodular row over [Formula: see text] of length [Formula: see text] can be mapped to a factorial row by elementary transformations.

Nice group structure on the elementary orbit space of unimodular rows

(1) If $R$ is an affine algebra of dimension $d\geq 4$ over $\overline{\mathbb{F}}_{p}$ with $p>3$, then the group structure on ${\rm Um}_d(R)/{\rm E}_d(R)$ is nice. (2) If $R$ is a commutative noetherian ring of dimension $d\geq 2$ such that ${\rm E}_{d+1}(R)$ acts transitively on ${\rm Um}_{d+1}(R),$ then the group structure on ${\rm Um}_{d+1}(R[X])/{\rm E}_{d+1}(R[X])$ is nice.

A note on cancellation of projective modules

We prove the following results. (i) Let A be an affine algebra of dimension d ≥ 4 over F ¯ p (with p ≥ d ). Then all projective A -modules of rank d − 1 are cancellative. (ii) Let A be a ring of dimension d such that E d + 1 ( R ) acts transitively on Um d + 1 ( R ) for every finite extension R of A . Then for any projective A -module P of rank d , E ( A ⊕ P ) acts transitively on Um ( A ⊕ P ) .

A stably elementary homotopy

If R is an affine algebra of dimension d over a perfect C 1 field and σ ∈ SL d+1 (R) is a stably elementary matrix, we show that there is a stably elementary matrix σ(X) ∈ SL d+1 (R[X]) with σ(1) = σ and σ(0) = I d+1 .

Cancellation problem for projective modules over affine algebras

AbstractLet A be an affine algebra of dimension n over an algebraically closed field k with 1/n! ∈ k. Let P be a projective A-module of rank n − 1. Then, it is an open question due to N. Mohan Kumar, whether P is cancellative. We prove the following results:(i) If A = R[T,T−1], then P is cancellative.(ii) If A = R[T,1/f] or A = R[T,f1/f,…,fr/f], where f(T) is a monic polynomial and f,f1,…,fr is R[T]-regular sequence, then An−1 is cancellative. Further, if k = p, then P is cancellative.

A cancellation theorem for projective modules over affine algebras over C1-fields

Let k be a C 1-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 A d 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.

Stably Free Modules

In [Su. Prob. 3], Suslin had asked the following question: Let A be any affine algebra of dimension n over an algebraically closed field. What is the smallest integer m such that all stably free projective modules of rank bigger than m are free? All the examples in the literature of stably free non-free modules have rank less than or equal to (n 1)/2. The aim of this note is to construct examples of such modules of large rank. We construct rank p stably free non-free modules over (p + 2)-dimensional affine algebras over algebraically closed fields, wherep is any prime. These varieties are actually smooth and rational. Over C, these are trivial as holomorphic vector bundles. [Forp > 2, this is classical. Forp = 2, see [MS]]. So these are strictly algebraic examples. I had described this construction in [MK 1] and proved the result for p = 2. We will reproduce the construction with necessary modifications in this note. Let p be any prime number and k any field. Letf (x) be any polynomial of degree p over k. Letf (0) = a E k* and Fi (xo, xl) = F(xo,x1) = xP *f (x0 /x1 ). Also let

ON STABLY FREE MODULES

In this paper we show that if A is an affine algebra of dimension n over an algebraically closed field, then each stably free module whose rank is greater than or equal to n is free. We also obtain some results on orbits of unimodular rows.Bibliography: 17 titles.

On the Homological Dimension of Algebras of Differential Operators

Let A be a commutative algebra over a field k, and VA be the k-subalgebra of Endk(A) generated by EndA(A) = A and all k-derivations of A. A study of the homological properties of VA was initiated by Hochschild, Kostant, and Rosenberg in [5], and continued by Rinehart [8], [9], Roos [11], Bjork [1], Rinehart and Rosenberg [10], and others. It was proved in [5] that, if k is perfect and A is a regular affine algebra of dimension r, then the global dimension of VA is between r and 2r. Moreover, if k has positive characteristic, then gl.dim VA = 2r [8]. By a recent celebrated theorem of Roos [11], gl.dim VA = r if k has characteristic zero and A = k[x1, …, xr]; in this case VA is the so-called “Weyl algebra on 2r variables”.