DOI

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank ≥ n, if every normal semisimple reductive R-subgroup of G contains (Gm,R)n. We prove that if G has isotropic rank ≥ 1 and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra A= R[x 1 , … , x n ] / I over R, the map HNis1(A,G)→HNis1(R,G) induced by evaluation at x 1 = ⋯ = x n = 0 , is a bijection. If k has characteristic 0, then, moreover, the map He´t1(A,G)→He´t1(R,G) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is ≥ 2 , and A is square-free, then K1G(A)=K1G(R), where K1G(R)=G(R)/E(R) is the corresponding non-stable K 1 -functor, also called the Whitehead group of G. The corresponding statements for G= GL n were previously proved by Ton Vorst.

Язык оригиналаанглийский
Страницы (с-по)509-524
Число страниц16
ЖурналJournal of Homotopy and Related Structures
Том14
Номер выпуска2
DOI
СостояниеОпубликовано - 11 июн 2019

    Предметные области Scopus

  • Геометрия и топология
  • Алгебра и теория чисел

ID: 43423347