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

Abstract

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.

The Pillars of Relative Quillen–Suslin Theory

“…This argument works with the Noetherian excision ring R⊕I rather than the use of the (non-Noetherian) Excision ring Z ⊕ I, and the Excision theorem of W. van der Kallen in [15], as is commonly used. We refer [11] to see other interesting applications of the Noetherian Excision rings.…”

Projective and stably free modules over real affine algebras

“…The purpose of this section is to state and prove (Theorem 1.15) a relative version of Theorem 1.7 which will be needed to improve the injective stability of SK 1 in section 3. We will began by recalling an interesting fact about the excision ring (see [21], Definition 2.5). For any ideal I ⊂ A, the excision ring A ⊕ I can be viewed as a fiber product of A with respect to the ideal I.…”

Nice group structure on the elementary orbit space of unimodular rows

“…(3) If R is a smooth affine algebra of dimension d ≥ 3 over an algebraically closed field, then Gupta, Garge and Rao [7] proved that the group structure on Um d (R)/E d (R) is nice. (4) If R is a local ring of dimension d with 2R = R, then Garge and Rao [6] proved that the group structure on Um…”