A module structure on certain orbit sets of unimodular rows

Kallen, Wilberd van der

doi:10.1016/0022-4049(89)90035-2

Cited by 85 publications

(35 citation statements)

References 15 publications

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“…In the papers [12], [13] W. van der Kallen shows that the orbit space of unimodular rows modulo the elementary action Um n (A)/E n (A) has an abelian group structure if 2 ≤ d ≤ 2n − 4, where d is the stable dimension of A. In fact he defines the notion of a weak Mennicke n-symbol in [13] and shows that the orbit space is bijective to the universal weak Mennicke n-symbol WMS n (A), under the above conditions.…”

Section: Nice Group Structuresmentioning

confidence: 99%

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Some remarks on symplectic injective stability

Basu

Chattopadhyay

Rao

2010

Proc. Amer. Math. Soc.

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Abstract. It is shown that if A is an affine algebra of odd dimension d over an infinite field of cohomological dimension at most one, with (d + 1)!A = A, and with 4|(d − 1), then Um d+1 (A) = e 1 Sp d+1 (A). As a consequence it is shown that if A is a non-singular affine algebra of dimension d over an infinite field of cohomological dimension at most one, and d!A = A, and 4|d, then. This result is a partial analogue for evendimensional algebras of the one obtained by Basu and Rao for odd-dimensional algebras earlier.

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Section: Nice Group Structuresmentioning

confidence: 99%

“…In fact he defines the notion of a weak Mennicke n-symbol in [13] and shows that the orbit space is bijective to the universal weak Mennicke n-symbol WMS n (A), under the above conditions.…”

Section: Nice Group Structuresmentioning

confidence: 99%

Some remarks on symplectic injective stability

Basu

Chattopadhyay

Rao

2010

Proc. Amer. Math. Soc.

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“…Using this we deduce in Theorem 3.6 that if I is an ideal in a ring R, and the orbit spaces MSE n (R), and MSE n (R, I) have the usual group structures (see [25,26]), then the group structure on MSE n (R, I) is nice (i.e. is Mennicke-like) if it is nice for the Excision ring MSE n (R ⊕ I).…”

Section: Introductionmentioning

confidence: 99%

“…(See Theorem 2.6). Later in [26] he showed that these orbit spaces also have a group structure when the size is a bit beyond half the dimension (the so-called Borsuk estimate).…”

Section: Introductionmentioning

confidence: 99%

A nice group structure on the orbit space of unimodular rows-II

2014

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Abstract. We establish an Excision type theorem for niceness of group structure on the orbit space of unimodular rows of length n modulo elementary action. This permits us to establish niceness for relative versions of results for the cases when n = d + 1; d being the dimension of the base algebra. We then study and establish niceness for the case when n = d, and also establish a relative version, when the base ring is a smooth affine algebra over an algebraically closed field.

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“…In [19,20] W. van der Suslin matrix S r (v, w) (recalled in the preliminaries), is unclear and their properties unknown.…”

Section: Ln Vaserstein Established Inmentioning

confidence: 99%