Nice group structure on the elementary orbit space of unimodular rows

Unimodular rows over monoid extensions of overrings of polynomial rings

Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ a commutative cancellative torsion-free seminormal monoid. Then (1) Let $A$ be a ring of type $R[d,m,n]$ and $P$ be a projective $A[M]$-module of rank $r \geq max\{2,d+1\}$. Then the action of $E(A[M] \oplus P)$ on $Um(A[M] \oplus P)$ is transitive and (2) Assume $(R, m, K)$ is a regular local ring containing a field $k$ such that either $char$ $k=0$ or $ char$ $k = p$ and $tr$-$deg$ $K/\mathbb{F}_p \geq 1$. Let $A$ be a ring of type $R[d,m,n]^*$ and $f\in R$ be a regular parameter. Then all finitely generated projective modules…

经典群在幺半群环上幺模行的初等作用

Elementary Action of Classical Groups on Unimodular Rows Over Monoid Rings

本文研究经典群在幺半群环幺模行上的初等作用传递性，给出辛群与正交群情形下的维数界，推导出经典群K₁的满稳定化界，推广了J. Gubeladze的线性群相关结果。

Swan的Bertini应用到单模行

Applications of Swan's Bertini to unimodular rows

研究单模行的结构，探讨在不同条件下的群结构和唯一可除性，主要贡献是证明在特定条件下单模行的性质。

广义Mennicke-Newman引理

Generalised Mennicke-Newman lemma

本文推广Mennicke-Newman引理至m×n右可逆矩阵，用于研究交换noetherian环中unimodular行的轨道集群结构。

Euler cycles and Mennicke symbols

Let $R$ be a smooth affine domain of dimension $d\geq 2$ over an infinite perfect field $k$. We establish a morphism from the Euler class group $E^d(R)$ to $Um_{d+1}(R)/E_{d+1}(R)$, the group of elementary orbits of unimodular rows.

Algebraic vector bundles on spheres

We determine the first non-stable ${\mathbb A}^1$-homotopy sheaf of $SL_n$. Using techniques of obstruction theory involving the ${\mathbb A}^1$-Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank $\geq d-1$ on split smooth affine quadrics of dimension $2d-1$. These computations allow us to answer a question posed by Nori, which gives a criterion for completability of certain unimodular rows. Furthermore, we study compatibility of our computations of ${\mathbb A}^1$-homotopy sheaves with real and complex realization.

Quillen-Suslin theory for a structure theorem for the Elementary Symplectic Group

A new set of elementary symplectic elements is described, It is shown that these also generate the elementary symplectic group {\rm ESp}$_{2n}(R)$. These generators are more symmetrical than the usual ones, and are useful to study the action of the elementary symplectic group on unimodular rows. Also, an alternate proof of, {\rm ESp}$_{2n}(R)$ is a normal subgroup of {\rm Sp}$_{2n}(R)$, is shown using the Local Global Principle of D. Quillen for the new set of generators.

Cancellation of vector bundles of rank $3$ with trivial Chern classes on smooth affine fourfolds

If $n \equiv 0,1~mod~4$, we prove a sum formula $V_{θ_{0}} (a_{0},a_{R}^{n}) = n \cdot V_{θ_{0}} (a_{0},a_{R})$ for the generalized Vaserstein symbol whenever $R$ is a smooth affine algebra over a perfect field $k$ with $char(k) \neq 2$ such that $-1 \in {k^{\times}}^{2}$. This enables us to generalize a result of Fasel-Rao-Swan on transformations of unimodular rows via elementary matrices over normal affine algebras of dimension $d \geq 4$ over algebraically closed fields of characteristic $\neq 2$. As a consequence, we prove that any projective module of rank $3$ with trivial Chern classes o…

Finitely generated modules over quasi-Euclidean rings

Let R be a unital commutative ring and let $M$ be an $R$-module that is generated by $k$ elements but not less. Let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by the elementary matrices. In this paper we study the action of $E_n(R)$ by matrix multiplication on the set $Um_n(M)$ of unimodular rows of $M$ of length $n \ge k$. Assuming $R$ is moreover Noetherian and quasi-Euclidean, e.g., $R$ is a direct sum of finitely many Euclidean rings, we show that this action is transitive if $n > k$. We also prove that $Um_k(M) /E_k(R)$ is equipotent with the unit group of $R/(a_1)$ where $(a_1)$ is …