15 research outputs found
Results on of general quadratic groups
No full textIn the first part of this article we discuss the relative cases of
Quillen-Suslin's local-global principle for the general quadratic (Bak's
unitary) groups, and its applications for the (relative) stable and unstable
-groups. The second part is dedicated to the graded version of
the local-global principle for the general quadratic groups and its application
to deduce a result for Bass' nil groups.Comment: 17 pages. arXiv admin note: substantial text overlap with
arXiv:2101.0702
Local-Global Principle for Transvection Groups
No full textBak A, Basu R, Rao RA. Local-Global Principle for Transvection Groups. Proceedings of the American Mathematical Society. 2010;138(4):1191-1204.In this article we extend the validity of Suslin's Local-Global Principle for the elementary transvection subgroup of the general linear group GL(n)(R), the symplectic group SP2n(R), and the orthogonal group O-2n(R), where n > 2, to a Local-Global Principle for the elementary transvection subgroup of the automorphism group Aut(P) of either a projective module P of global rank > 0 and constant local rank > 2, or of a nonsingular symplectic or orthogonal module P of global hyperbolic rank > 0 and constant local hyperbolic rank > 2. In Suslin's results, the local and global ranks are the same, because he is concerned only with free modules. Our assumption that the global (hyperbolic) rank > 0 is used to define the elementary transvection subgroups. We show further that the elementary transvection subgroup ET(P) is normal in Aut(P), that ET(P) = T(P), where the latter denotes the full transvection subgroup of Aut(P), and that the unstable K-1-group K-1(Aut(P)) = Aut(P)/ET(P) = Aut(P)/T(P) is nilpotent by abelian, provided R has finite stable dimension. The last result extends previous ones of Bak and Hazrat for CLn(R), SP2n(R), and O-2n(R). An important application to the results in the current paper can be found in a preprint of Basu and Rao in which the last two named authors studied the decrease in the injective stabilization of classical modules over a nonsingular affine algebra over perfect C-1-fields. We refer the reader to that article for more details
Injective stability for K1 of classical modules
No full textAbstractIn Rao (1994) [14], the second author and W. van der Kallen showed that the injective stabilization bound for K1 of the general linear group is d+1 over a regular affine algebra over a perfect C1-field, where d is the Krull dimension of the base ring which is finite and at least 2. In this article we prove that the injective stabilization bound for K1 of the symplectic group is d+1 over a geometrically regular ring containing a field, where d is the dimension of the base ring which is finite and at least 2. Using the Local–Global Principle for the transvection subgroup of the automorphism group of projective and symplectic modules we show that the injective stabilization bound is d+1 for K1 of projective and symplectic modules of global rank at least 1 and local rank at least 3 respectively in each of the two cases above
A note on general quadratic groups
No full text We deduce an analogue of Quillen–Suslin’s local-global principle for the transvection subgroups of the general quadratic (Bak’s unitary) groups. As an application, we revisit the result of Bak–Petrov–Tang on injective stabilization for the [Formula: see text]-functor of the general quadratic groups. </jats:p
ABSENCE OF TORSION FOR <font>NK</font><sub>1</sub>(R) OVER ASSOCIATIVE RINGS
No full text When R is a commutative ring with identity, and if k ∈ ℕ, with kR = R, then it was shown in [C. Weibel, Mayer–Vietoris Sequence and Module Structure on NK0, Lecture Notes in Mathematics, Vol. 854 (Springer, 1981), pp. 466–498] that SK 1(R[X]) has no k-torsion. We prove this result for any associative ring R with identity in which kR = R. </jats:p
Injective stability for K1 of the orthogonal group
No full textAbstractThe injective stabilization bound of L.N. Vaserstein for K1 of the orthogonal group over an affine algebra over a perfect C1-field is discussed
