In 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 $\mathrm{K}_1$-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:2101.07022

• 11E57 - Classical groups
• 13A02 - Graded rings
• 13C10 - Projective and free modules and ideals in commutative rings
• 15A63 - Quadratic and bilinear forms, inner products
• 19B14 - Stability for linear groups
• 19D35 - Negative \(K\)-theory, NK and Nil
• 20F18 - Nilpotent groups

• 1 zbMATH Open