TY - JOUR

T1 - Isotropic reductive groups over discrete Hodge algebras

AU - Stavrova, Anastasia

PY - 2019/6/11

Y1 - 2019/6/11

N2 - 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.

AB - 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.

KW - Bass-Quillen conjecture

KW - discrete Hodge algebra

KW - G-torsor

KW - Milnor square

KW - non-stable K -functor

KW - reductive group

KW - simple algebraic group

KW - Stanley-Reisner ring

KW - Whitehead group

KW - non-stable K-1-functor

UR - http://www.scopus.com/inward/record.url?scp=85065559100&partnerID=8YFLogxK

U2 - 10.1007/s40062-018-0221-7

DO - 10.1007/s40062-018-0221-7

M3 - Article

AN - SCOPUS:85065559100

VL - 14

SP - 509

EP - 524

JO - Journal of Homotopy and Related Structures

JF - Journal of Homotopy and Related Structures

SN - 2193-8407

IS - 2

ER -