Unimodular Rows Research Articles

Elementary Action of Classical Groups on Unimodular Rows Over Monoid Rings

Elementary Action of Classical Groups on Unimodular Rows Over Monoid Rings

Motivic stable cohomotopy and unimodular rows

We relate the group structure of van der Kallen on orbit sets of unimodular rows ([26]) with values in a smooth algebra A over a field k with the motivic cohomotopy groups of X=SpecA with coefficients in An∖0 in the sense of [5]. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by An+1∖0 or, equivalently, by an A1-weakly equivalent quadric Q2n+1 to that considered in [5], defined by a quadric Q2n, by means of explicit morphisms Q2n+1→Q2n, Q2n×Gm→Q2n+1 of quadrics.

Suslin matrices and the Euler characteristic

Jose-Rao proved the Key Lemma for the action of the elementary matrices En(R) on the unimodular rows Umn(R) of length n. It essentially describes how Sn−1(vε,wεt−1) looks in terms of Sn−1(v,w), n≥3, when ε is an elementary generator Eij(λ) of En(R), and v,w∈Umn(R), with 〈v,w〉=1. In this paper, we prove the generalized Key Lemma for the action of SLn(R) on Umn(R), n≥3, which is inspired by the (unpublished) results of Fossum–Foxby–Iversen.

Embedding of Unimodular Row Vectors

In this paper, we mainly study the embedding problem of unimodular row vectors, focusing on avoiding the identification of polynomial zeros. We investigate the existence of the minimal syzygy module of the ZLP polynomial matrix and demonstrate that the minimal syzygy module has structural properties that are similar to the fundamental solution system of homogeneous linear equations found in linear algebra. Finally, we provide several embedding methods for unimodular vectors in certain cases.

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

This article concerns a question asked by M. V. Nori on homotopy of sections of projective modules defined on the polynomial algebra over a smooth affine domain R. While this question has an affirmative answer, it is known that the assertion does not hold if: (1) dim(R)=2; or (2) d≥3 but R is not smooth. We first prove that an affirmative answer can be given for dim(R)=2 when R is an F‾p-algebra. Next, for d≥3 we find the precise obstruction for the failure in the singular case. Further, we improve a result of Mandal (related to Nori's question) in the case when the ring A is an affine F‾p-algebra of dimension d. We apply this improvement to define the n-th Euler class group En(A), where 2n≥d+2. Moreover, if A is smooth, we associate to a unimodular row v of length n+1 its Euler class e(v)∈En(A) and show that the corresponding stably free module, say, P(v) has a unimodular element if and only if e(v) vanishes in En(A).

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 Completions and Orthogonal Complements of Matrices over Univariate Ore Extensions

We generalize an efficient hyper-regularity and unimodularity test from differential Ore polynomial matrices to arbitrary Ore polynomial matrices. The core of the contribution consists of algorithms for unimodular row and column completions of arbitrary univariate Ore polynomial matrices, respectively. After a possible degree reduction using noncommutative companion matrices, this is done by a systematic projection of suitable coordinates with a subsequent elimination. Based on previous work for differential Ore polynomial matrices, we remove rather restrictive conditions, which now allows for the application of this algorithm to a larger class of systems that may contain pure algebraic equations.

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≥max {2,d+1}. Then the action of E(A[M]⊕P) on (A[M]⊕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∕𝔽p≥1. Let A be a ring of type R[d,m,n]∗ and f∈R be a regular parameter. Then all finitely generated projective modules over A[M], A[M]f and A[M]⊗RR(T) are free. When M is free both results are due to Keshari and Lokhande (2014).

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.

GE2-rings and a graph of unimodular rows

For a commutative ring A we consider a related graph, Γ ( A ) , whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that Γ ( A ) is path-connected if and only if A is a GE 2 -ring, in the terminology of P. M. Cohn. Furthermore, if Y ( A ) denotes the clique complex of Γ ( A ) , we prove that Y ( A ) is simply connected if and only if A is universal for GE 2 . More precisely, our main theorem is that for any commutative ring A the fundamental group of Y ( A ) is isomorphic to the group K 2 ( 2 , A ) modulo the subgroup generated by symbols.

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

If n ≡ 0 , 1 m o d 4 , we prove a sum formula V θ 0 ( a 0 , a R n ) = n ⋅ 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 c h a r ( k ) ≠ 2 such that − 1 ∈ k × 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 ≥ 4 over algebraically closed fields of characteristic ≠2. As a consequence, we prove that any projective module of rank 3 with trivial Chern classes over a smooth affine algebra of dimension 4 over an algebraically closed field k with c h a r ( k ) ≠ 2 , 3 is cancellative.

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.

Parametric Solutions of System of Linear Diophantine Equations by Crushing Method

There are studies on parametric solutions of system of Linear Diophantine equations based on uni-modular reductions of the coefficient matrix. In this paper we generate parametric solutions, with uni-modular row reductions on the coefficient matrix, based on the steps used in obtaining gcd of the coefficients in a row by crushing method. This application of gcd by crushing specifies an order for the row reductions and enables to give algorithm for the computations.

Two lives: Compositions of unimodular rows

The paper lays the foundation for the study of unimodular rows using Spin groups. We show that elementary orbits of unimodular rows (of any length n≥3) are equivalent to elementary Spin orbits on the unit sphere. (This bijection is proved over all commutative rings.) In the special case n=3, this leads to an interpretation of the Vaserstein symbol using Spin groups.In addition, we introduce a new composition law that operates on certain subspaces of the underlying quadratic space (using the multiplication in composition algebras). In particular, the special case of split-quaternions leads to the composition of unimodular rows (discovered by L. Vaserstein and later generalized by W. van der Kallen). Strikingly, with this approach, we now see the possibility of new orbit structures not only for unimodular rows (using octonion multiplication) but also for more general quadratic spaces.

Generalised Mennicke–Newman lemma

Mennicke–Newman lemma for unimodular rows was used by W. van der Kallen to give a group structure on the orbit set \(\frac{Um_{n}(R)}{E_{n}(R)}\) for a commutative noetherian ring of dimension \(d\le 2n-4.\) In this paper, we generalise the Mennicke–Newman lemma for \(m\times n \) right invertible matrices

Completion of skew completable unimodular rows

In this paper, we prove that if R is a local ring of dimension \(d\ge 3\), d odd and \(\frac{1}{(d-1)!}\in R\) then any skew completable unimodular row \(v\in Um_{d}(R[X])\) is completable. It is also proved that skew completable unimodular rows of size \(d\ge 3\) over a regular local ring of dimension d are first row of a 2- stably elementary matrix.

On completion of unimodular rows over polynomial extension of finitely generated rings over [formula omitted

In this article, we prove that if R is a finitely generated ring over Z of dimension d,d≥2,1d!∈R, then any unimodular row over R[X] of length d+1 can be mapped to a factorial row by elementary transformations.

GLOBAL ACTIONS AND VECTOR -THEORY

Purely algebraic objects like abstract groups, coset spaces, and G-modules do not have a notion of hole as do analytical and topological objects. However, equipping an algebraic object with a global action reveals holes in it and thanks to the homotopy theory of global actions, the holes can be described and quantified much as they are in the homotopy theory of topological spaces. Part I of this article, due to the first author, starts by recalling the notion of a global action and describes in detail the global actions attached to the general linear, elementary, and Steinberg groups. With these examples in mind, we describe the elementary homotopy theory of arbitrary global actions, construct their homotopy groups, and revisit their covering theory. We then equip the set $Um_{n}(R)$ of all unimodular row vectors of length $n$ over a ring $R$ with a global action. Its homotopy groups $\unicode[STIX]{x1D70B}_{i}(Um_{n}(R)),i\geqslant 0$ are christened the vector $K$ -theory groups $K_{i+1}(Um_{n}(R)),i\geqslant 0$ of $Um_{n}(R)$ . It is known that the homotopy groups $\unicode[STIX]{x1D70B}_{i}(\text{GL}_{n}(R))$ of the general linear group $\text{GL}_{n}(R)$ viewed as a global action are the Volodin $K$ -theory groups $K_{i+1,n}(R)$ . The main result of Part I is an algebraic construction of the simply connected covering map $\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R)$ where $\mathit{EUm}_{n}(R)$ is the path connected component of the vector $(1,0,\ldots ,0)\in Um_{n}(R)$ . The result constructs the map as a specific quotient of the simply connected covering map $St_{n}(R)\rightarrow E_{n}(R)$ of the elementary global action $E_{n}(R)$ by the Steinberg global action $St_{n}(R)$ . As expected, $K_{2}(Um_{n}(R))$ is identified with $\text{Ker}(\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R))$ . Part II of the paper provides an exact sequence relating stability for the Volodin $K$ -theory groups $K_{1,n}(R)$ and $K_{2,n}(R)$ to vector $K$ -theory groups.

Variations in 𝔸1 on a Theme of Mohan Kumar

Abstract For every prime $p$, Mohan Kumar constructed examples of stably free modules of rank $p$ on suitable $p+1$-dimensional smooth affine varieties. This note discusses how to detect the corresponding unimodular rows by an explicit motivic cohomology group. Using the recent developments in the $\mathbb{A}^1$-obstruction classification of vector bundles, this provides an alternative proof of non-triviality of Mohan Kumar’s stably free modules. The reinterpretation of Mohan Kumar’s examples also allows to produce interesting examples of stably trivial torsors for other algebraic groups.

Odd orthogonal matrices and the non-injectivity of the Vaserstein symbol

R.A. Rao–W. van der Kallen showed that the Vaserstein symbol VΓ(SR3) from the orbit space of unimodular rows of length three over the coordinate ring of the real three sphere SR3 modulo elementary action to the elementary symplectic Witt group WE(Γ(SR3)) is not injective. Dhvanita R. Rao–Neena Gupta gave an uncountable family of singular real threefolds Aα for which the Vaserstein symbol VAα is not injective. In this paper, we give a countable family of smooth real birationally equivalent threefolds An for which the Vaserstein symbol VAn is not injective.