ON the CONGRUENCE KERNEL of ISOTROPIC GROUPS over RINGS

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ON the CONGRUENCE KERNEL of ISOTROPIC GROUPS over RINGS. / Stavrova, Anastasia.

In: Transactions of the American Mathematical Society, Vol. 373, No. 7, 07.2020, p. 4585-4626.

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Stavrova, Anastasia. / ON the CONGRUENCE KERNEL of ISOTROPIC GROUPS over RINGS. In: Transactions of the American Mathematical Society. 2020 ; Vol. 373, No. 7. pp. 4585-4626.

BibTeX

@article{1301c652bba146a2b6d9c121664aed31,

title = "ON the CONGRUENCE KERNEL of ISOTROPIC GROUPS over RINGS",

abstract = "Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ≥ 2. The elementary subgroup E(R) of G(R) is the subgroup generated by UP+(R) and UP−(R), where UP± are the unipotent radicals of two opposite parabolic subgroups P± of G. Assume that 2 ∈ R× if G is of type Bn, Cn, F4, G2 and 3 ∈ R× if G is of type G2. We prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism E(R) → E(R) between the profinite completion of E(R) and the congruence completion of E(R) with respect to congruence subgroups of finite index, is central in E(R). In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of E(R) if R is a local ring.",

keywords = "Congruence kernel, Congruence subgroup problem, Elementary subgroup, Isotropic reductive group, Steinberg group",

author = "Anastasia Stavrova",

year = "2020",

month = jul,

doi = "10.1090/tran/8091",

language = "English",

volume = "373",

pages = "4585--4626",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "7",

}

RIS

TY - JOUR

T1 - ON the CONGRUENCE KERNEL of ISOTROPIC GROUPS over RINGS

AU - Stavrova, Anastasia

PY - 2020/7

Y1 - 2020/7

N2 - Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ≥ 2. The elementary subgroup E(R) of G(R) is the subgroup generated by UP+(R) and UP−(R), where UP± are the unipotent radicals of two opposite parabolic subgroups P± of G. Assume that 2 ∈ R× if G is of type Bn, Cn, F4, G2 and 3 ∈ R× if G is of type G2. We prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism E(R) → E(R) between the profinite completion of E(R) and the congruence completion of E(R) with respect to congruence subgroups of finite index, is central in E(R). In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of E(R) if R is a local ring.

AB - Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ≥ 2. The elementary subgroup E(R) of G(R) is the subgroup generated by UP+(R) and UP−(R), where UP± are the unipotent radicals of two opposite parabolic subgroups P± of G. Assume that 2 ∈ R× if G is of type Bn, Cn, F4, G2 and 3 ∈ R× if G is of type G2. We prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism E(R) → E(R) between the profinite completion of E(R) and the congruence completion of E(R) with respect to congruence subgroups of finite index, is central in E(R). In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of E(R) if R is a local ring.

KW - Congruence kernel

KW - Congruence subgroup problem

KW - Elementary subgroup

KW - Isotropic reductive group

KW - Steinberg group

UR - http://www.scopus.com/inward/record.url?scp=85087130397&partnerID=8YFLogxK

U2 - 10.1090/tran/8091

DO - 10.1090/tran/8091

M3 - Article

VL - 373

SP - 4585

EP - 4626

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 7

ER -

ID: 5758788