We establish a relative version of the abstract “affine representability” theorem in A 1–
homotopy theory from part I of this paper. We then prove some A 1–invariance statements for …

In the current article we study the structure of a Chevalley group G (R) over a commutative
ring R. We generalize and improve the following results on:• the standard, relative, and multi …

This paper studies the work of Bak in algebra and (lower) algebraic K-theory and some later
developments stimulated by them. We present an overview of his work in these areas …

The present paper is the [slightly expanded] text of our talk at the Conference" Advances in
Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert that …

Let G be a reductive algebraic group over a field k, such that every semisimple normal
subgroup of G has isotropic rank≥ 2, ie contains (Gm) 2. Let K1G be the non-stable K1 …

Abstract Let G= G (Φ, R) be the simply connected Chevalley group with root system Φ over a
ring R. Denote by E (Φ, R) its elementary subgroup. The main result of the article asserts that …

In the present paper, we discuss some recent versions of localization methods for
calculations in the groups of points of algebraic-like and classical-like groups. Namely, we …

Let A be an algebra which is a direct limit of module finite algebras over a commutative ring
R with 1. Let I, J be two-sided ideals of A, GL n (A, I) the principal congruence subgroup of …

We revisit localisation and patching method in the setting of Chevalley groups. Introducing
certain subgroups of relative elementary Chevalley groups, we develop relative versions of …

Let R be an associative ring with 1; A, B⊴ R be its ideals; C (n, R, A) be the full congruence
subgroup of level A in GL (n, R); and E (n, R, A) be the relative elementary subgroup of level …